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Algebraic Logic, Quantum Algebra and Algebraic Mathematics PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Sun, 16 Jan 2011 23:37:54 UTC Contents Articles Algebraic Logic, Quantum Logic, Quantum Algebra, Algebra, Algebraic Geometry, Algebraic Topology, Category Theory and Higher Dimensional Algebra v.2min ! Boolean logic 1 Intuitionistic logic 7 Hey ting arithmetic 13 Algebraic Logic and Many- Valued Logic 14 Algebraic logic 14 Lukasiewicz logic 16 Ternary logic 18 Multi-valued logic 21 Mathematical logic 24 Symbolic logic 37 Metalogic 37 Metatheory 40 Metamathematics 41 Abstract Algebra 43 Abstract algebra 43 Universal algebra 46 Heyting algebra 50 MV-algebra 58 Group theory 60 Abelian group 68 Group algebra 75 Cayley's theorem 77 80 Special Algebras, Operator Algebra and Quantum Algebra Lie algebra 80 Lie group 85 Affine Lie algebra 94 Kac— Moody algebra 96 Hopf algebra 99 Quantum group 104 Affine quantum group 111 Group representation 112 Unitary representation 115 Representation theory of the Lorentz group 117 Stone— von Neumann theorem 121 Peter— Weyl theorem 126 Quantum algebra 128 Quantum affine algebra 129 Clifford algebra 130 Von Neumann algebra 140 C*-algebra 150 Quasi-Hopf algebra 155 Quasitriangular Hopf algebra 156 Ribbon Hopf algebra 157 Quasi-triangular Quasi-Hopf algebra 158 Quantum inverse scattering method 159 Grassmann algebra 160 Supergroup 174 Superalgebra 174 Algebroid 177 Algebraic Geometry 178 Algebraic geometry 178 List of algebraic geometry topics 186 Duality (projective geometry) 191 Universal algebraic geometry 197 Motive (algebraic geometry) 197 Grothendieck— Hirzebruch— Riemann— Roch theorem 204 Coherent sheaf 206 Grothendieck topology 208 Crystalline cohomology 214 De Rham cohomology 218 Algebraic geometry and analytic geometry 22 1 Riemannian manifold 224 List of complex analysis topics 229 Algebraic Topology and Groupoids 232 Algebraic topology 232 Groupoid 236 Galois group 241 Grothendieck group 242 Esquisse d'un Programme 245 Galois theory 247 Grothendieck' s Galois theory 253 Galois cohomology 254 Homological algebra 255 Homology theory 259 Homotopical algebra 262 Cohomology theory 263 K-theory 266 Algebraic K-theory 268 Topological K-theory 273 Category Theories 275 Category theory 275 Category (mathematics) 282 Glossary of category theory 287 Dual (category theory) 289 Abelian category 290 Yoneda lemma 293 Limit (category theory) 296 Adjoint functors 305 Natural transformations 319 Algebraic category 322 Domain theory 324 Enriched category theory 328 Topos 331 Descent (category theory) 336 Stack (descent theory) 338 Categorical logic 339 Timeline of category theory and related mathematics 342 List of important publications in mathematics 360 Higher Dimensional Algebras (HDA) 383 Higher-dimensional algebra 383 Higher category theory 387 Duality (mathematics) 389 References Article Sources and Contributors 398 Image Sources, Licenses and Contributors 403 Article Licenses License 404 Algebraic Logic, Quantum Logic, Quantum Algebra, Algebra, Algebraic Geometry, Algebraic Topology, Category Theory and Higher Dimensional Algebra v.2min Boolean logic Boolean logic is a complete system for logical operations, used in many systems. It was named after George Boole, who first defined an algebraic system of logic in the mid 19th century. Boolean logic has many applications in electronics, computer hardware and software, and is the basis of all modern digital electronics. In 1938, Claude Shannon showed how electric circuits with relays could be modeled with Boolean logic. This fact soon proved enormously consequential with the emergence of the electronic computer. Using the algebra of sets, this article contains a basic introduction to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications. The Boolean algebra (structure) article discusses a type of algebraic structure that satisfies the axioms of Boolean logic. The binary arithmetic article discusses the use of binary numbers in computer systems. Set logic vs. Boolean logic Sets can contain any elements. We will first start out by discussing general set logic, then restrict ourselves to Boolean logic, where elements (or "bits") each contain only two possible values, called various names, such as "true" and "false", "yes" and "no", "on" and "off", or "1" and "0". Terms Boolean logic Let X be a set: • An element is one member of a set and is denoted by £ . If the element is not a member of a set it is denoted by d . • The universe is the set X, sometimes denoted by 1 . Note that this use of the word universe means "all elements being considered", which are not necessarily the same as "all elements there are ". • The empty set or null set is the set of no elements, denoted by and sometimes 0. • A unary operator applies to a single set. There is only one unary operator, called logical NOT. It works by taking the complement with respect to the universe, i.e. the set of all elements under consideration. • A binary operator applies to two sets. The basic binary operators are logical OR and logical AND. They perform the union and intersection of sets. There are also other derived binary operators, such as XOR (exclusive OR, i.e., "one or the other, but not both"). • A subset is denoted by A C B and means every element in set A is also in set B. • A superset is denoted by A ~D B and means every element in set B is also in set A. • The identity or equivalence of two sets is denoted by A = B an d means that every element in set A is also in set B and every element in set B is also in set A. • A proper subset is denoted by A C B an d means every element in set A is also in set B and the two sets are not identical. • A proper superset is denoted by A D B an d means every element in set B is also in set A and the two sets are not identical. Venn diagram showing the intersection of sets "A AND B" (in violet/dark shading), the union of sets "A OR B" (all the colored regions), and the exclusive OR case "set A XOR B" (all the colored regions except the violet). The "universe" is represented by all the area within the rectangular frame. Boolean logic Example Imagine that set A contains all even numbers (multiples of two) in "the universe" (defined in the example below as all integers between and 30 inclusive) and set B contains all multiples of three in "the universe". Then the intersection of the two sets (all elements in sets A AND B) would be all multiples of six in "the universe". The complement of set A (all elements NOT in set A) would be all odd numbers in "the universe". Chaining operations together UNIVERSE (Integers from to 30) 1 7 11 13 17 19 23 29 3ETA(Ev be ven numbers/Multiples of 2) 2 4 8 14 16 22 26 28 SET AC (Multiples _pf 10) SET C (Multiples of SET AB (Multiples of 6) 6 12 18 24 SET ABC (Multiples of 30) 30 SETBC (Multiples of 1 While at most two sets are joined in any Boolean operation, the new set formed by that operation can then be joined with other sets utilizing additional Boolean operations. Using the previous example, we can define a new set C as the set of all multiples of five in "the universe". Thus "sets A AND B AND C" would be all multiples of 30 in "the universe". If more convenient, we may consider set AB to be the intersection of sets A and B, or the set of all multiples of six in "the universe". Then we can say "sets AB AND C" are the set of all multiples of 30 in "the universe". We could then take it a step further, and call this result set ABC. Use of parentheses While any number of logical ANDs (or any number of logical ORs) may be chained together without ambiguity, the combination of ANDs and ORs and NOTs can lead to ambiguous cases. In such cases, parentheses may be used to clarify the order of operations. As always, the operations within the innermost pair is performed first, followed by the next pair out, etc., until all operations within parentheses have been completed. Then any operations outside the parentheses are performed. Application to binary values In this example we have used natural numbers, while in Boolean logic binary numbers are used. The universe, for example, could contain just two elements, "1" and "0" (or "true" and "false", "yes" and "no", "on" or "off", etc.). We could also combine binary values together to get binary words, such as, in the case of two digits, "00", "01", "10", and "11". Applying set logic to those values, we could have a set of all values where the first digit is "0" ("00" and "01") and the set of all values where the first and second digits are different ("01" and "10"). The intersection of the two sets would then be the single element, "01". This could be shown by the following Boolean expression, where "1st" is the first digit and "2nd" is the second digit: (NOT 1st) AND (1st XOR 2nd) Boolean logic Properties We define symbols for the two primary binary operations as A/fl (logical AND/set intersection) and V/U (logical OR/set union), and for the single unary operation — 1/ ~ (logical NOT/set complement). We will also use the values (logical FALSE/the empty set) and 1 (logical TRUE/the universe). The following properties apply to both Boolean logic and set logic (although only the notation for Boolean logic is displayed here): a V (6 V c) = = (a Vfo) V c a A (6 A c) : = (a Afo) A c associativity aVfo = foV a a A fo = fo A a commutativity a V (a A fo) : = a a A (a V fo) : = a absorption a V (6 A c) = = (a Vfo) A (a Vc) a A (fo V c) = = (a Afo) V (a Ac) distributivity a V -ia = 1 a A -ia = complements aV a = a a A a = a idempotency sVO = a a A 1 = a boundedness a VI = 1 aAO = -.0= 1 -.1 = and 1 are complements -.(a V b) = -■a ^ v-.6 -.(a A 6) = -.a V '-ib de Morgan's laws —•—•a = a involution The first three properties define a lattice; the first five define a Boolean algebra. The remaining five are a consequence of the first five. Other notations Mathematicians and engineers often use plus (+) for OR and a product sign ( ■ ) for AND. OR and AND are somewhat analogous to addition and multiplication in other algebraic structures, and this notation makes it very easy to get sum of products form for normal algebra. NOT may be represented by a line drawn above the expression being negated ( x )• It also commonly leads to giving ■ a higher precedence than +, removing the need for parenthesis in some cases. Programmers will often use a pipe symbol (I) for OR, an ampersand (&) for AND, and a tilde (~) for NOT. In many programming languages, these symbols stand for bitwise operations. "II", "&&", and "!" are used for variants of these operations. Another notation uses "meet" for AND and "join" for OR. However, this can lead to confusion, as the term "join" is also commonly used for any Boolean operation which combines sets together, which includes both AND and OR. Basic mathematics use of Boolean terms • In the case of simultaneous equations, they are connected with an implied logical AND: x + y = 2 AND x - y = 2 • The same applies to simultaneous inequalities: x + y < 2 AND x-y<2 Boolean logic • The greater than or equals sign ( > ) and less than or equals sign ( < ) may be assumed to contain a logical OR: X<2 OR X = 2 • The plus/minus sign ( ± ), as in the case of the solution to a square root problem, may be taken as logical OR: WIDTH = 3 OR WIDTH = -3 English language use of Boolean terms Care should be taken when converting an English sentence into a formal boolean statement. Many English sentences have imprecise meanings. • In certain cases, AND and OR can be used interchangeably in English: / always carry an umbrella for when it rains and snows has the same meaning as I always carry an umbrella for when it rains or snows. An alternate phrasing would be / always carry an umbrella for when precipitation is forecast. • Sometimes the English words "and" and "or" have a meaning that is apparently opposite of its meaning in boolean logic: "Give me all the red and blue berries" usually means, "Give me all the berries that are either red or blue". An alternative phrasing for this request would be, "Give me all berries that are red and all berries that are blue." • Depending on the context, the word "or" may correspond with either logical OR (which corresponds to the English equivalent "and/or") or logical XOR (which corresponds to the English equivalent "either/or"): • The waitress asked, "Would you like cream or sugar with your coffee?" This is an example of a "Logical OR", whereby the choices are cream, sugar, or cream and sugar (in addition to none of the above). • The waitress asked, "Would you like soup or salad with your meal?" This is an example of a "Logical XOR", whereby the choices are soup or salad (or neither), but soup and salad are not an option.) • This can be a significant challenge when providing precise specifications for a computer program or electronic circuit in English. The description of such functionality may be ambiguous. Take for example the statement, "The program should verify that the applicant has checked the male or female box." This is usually interpreted as an XOR and so a verification is performed to ensure that one, and only one, box is selected. In other cases the intended interpretation of English may be less obvious; the author of the specification should be consulted to determine the original intent. Applications Digital electronic circuit design Boolean logic is also used for circuit design in electrical engineering; here and 1 may represent the two different states of one bit in a digital circuit, typically high and low voltage. Circuits are described by expressions containing variables, and two such expressions are equal for all values of the variables if, and only if, the corresponding circuits have the same input-output behavior. Furthermore, every possible input-output behavior can be modeled by a suitable Boolean expression. Basic logic gates such as AND, OR, and NOT gates may be used alone, or in conjunction with NAND, NOR, and XOR gates, to control digital electronics and circuitry. Whether these gates are wired in series or parallel controls the precedence of the operations. Boolean logic Database applications Relational databases use SQL, or other database-specific languages, to perform queries, which may contain Boolean logic. For this application, each record in a table may be considered to be an "element" of a "set". For example, in SQL, these SELECT statements are used to retrieve data from tables in the database: SELECT * FROM employees WHERE last_name = 'Dean' AND first_name = ' James ' ; This example will produce a list of all employees, and only those employees, named James Dean. SELECT * FROM employees WHERE last_name = 'Dean' OR first_name = ' James ' ; This example will produce a list of all employees whose first name is James OR whose last name is Dean. Any and all employees named James Dean (from the first example) would also appear in this list. SELECT * FROM employees WHERE NOT last_name = 'Dean' ; This example will produce a list of all employees whose last name is not Dean. All employees named James from the second example would appear on this list, except for those employees named James Dean. Parentheses may be used to explicitly specify the order in which Boolean operations occur, when multiple operations are present: SELECT * FROM employees WHERE (NOT last_name = 'Smith') AND (first_name = 'John' OR first_name = 'Mary') ; This example will produce a list of employees named John OR named Mary, but specifically excluding those named John Smith or Mary Smith. Multiple sets of nested parentheses may also be used, where needed. Any Boolean operation (or operations) which combines two (or more) tables together is referred to as a join, in relational database terminology. In the field of Electronic Medical Records, some software applications use Boolean logic to query their patient databases, in what has been named Concept Processing technology. Search engine queries Search engine queries also employ Boolean logic. For this application, each web page on the Internet may be considered to be an "element" of a "set". The following examples use a syntax supported by Google. • Doublequotes are used to combine whitespace-separated words into a single search term. • Whitespace is used to specify logical AND, as it is the default operator for joining search terms: " Search term 1" "Search term 2" • The OR keyword is used for logical OR: " Search term 1" OR "Search term 2" • The minus si gn is used for logical NOT (AND NOT): " Search term 1" -"Search term 2" Boolean logic Notes and references [1] Not all search engines support the same query syntax. Additionally, some organizations provide "specialized" search engines that support alternate or extended syntax. (See e.g., Syntax Cheatsheet (http://www.google.com/help/cheatsheet.html), Google codesearch supports regular expressions (http://www.google.eom/intl/en/help/faq_codesearch.html#regexp)). [2] Doublequote-delimited search terms are called "exact phrase" searches in the Google documentation. External links • The Calculus of Logic (http://www.maths.tcd.ie/pub/HistMath/People/Boole/CalcLogic/CalcLogic.html), by George Boole, Cambridge and Dublin Mathematical Journal Vol. Ill (1848), pp. 183-98. • Logical Formula Evaluator (http://sourceforge.net/projects/logicaleval/) (for Windows), a software which calculates all possible values of a logical formula • Maiki & Boaz BDD-PROJECT (http://www.bdd-project.com), a Web Application for BDD reduction and visualization. Intuitionistic logic Intuitionistic logic, or constructive logic, is a symbolic logic system that differs from classical logic in its definition of what it means for a statement to be true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either. In constructive logic, a statement is only true if there is a proof that it is true, and only false if there is a proof that it is false. Operations in constructive logic preserve justification, rather than truth. Syntactically, intuitionist logic differs from classical logic in that the law of excluded middle and double negation elimination are not axioms of the system, and cannot be proved in it. Constructive logic is practically useful because its restrictions produce proofs that have the existence property, making it also suitable for other forms of mathematical constructivism. Informally, this means that if you have a constructive proof that an object exists, you can turn that constructive proof into an algorithm for generating an example of it. It was originally developed by Arend Heyting to provide a formal basis for Brouwer's programme of intuitionism. Intuitionistic logic Syntax The syntax of formulas of intuitionistic logic is similar to propositional logic or first-order logic. However, intuitionistic connectives are not definable in terms of each other in the same way as in classical logic, hence their choice matters. In intuitionistic propositional logic it is customary to use — >, a, v, J. as the basic connectives, treating ->A as an abbreviation for (A — > J.). In intuitionistic first-order logic both quantifiers 3, V are needed. (((^^p- p) - (pv -p)) - (-pv -.-. p)) v(H-.p-p)-(pv^p)) «— p - p) - (pv -p)) -.(-.pv-.- — pv(— p (((-.-.p- p) - (pv -ipB - (-.pv -.-.p)) - (-"PV (-'-'P - p» (-.-.p- p)v((-.-.p- p) - (pv^p)) (-•-■ P-P)-(pv-p) pv--.p The Rieger— Nishimura lattice. Its nodes are the propositional formulas in one variable up to intuitionistic logical equivalence, ordered by intuitionistic logical implication. Many tautologies of classical logic can no longer be proven within intuitionistic logic. Examples include not only the law of excluded middle p v -<p, but also Peirce's law ((p — » q) — > p) — > /?, and even double negation elimination. In classical logic, both p — > —1—1/7 and also —1—1/7 — > p are theorems. In intuitionistic logic, only the former is a theorem: double negation can be introduced, but it cannot be eliminated. Rejecting p v -77 may seem strange to those more familiar with classical logic, but proving this statement in constructive logic would require producing a proof for the truth or falsity of all possible statements, which is impossible for a variety of reasons. Because many classically valid tautologies are not theorems of intuitionistic logic, but all theorems of intuitionist logic are valid classically, intuitionist logic can be viewed as a weakening of classical logic, albeit one with many useful properties. Sequent calculus Gentzen discovered that a simple restriction of his system LK (his sequent calculus for classical logic) results in a system which is sound and complete with respect to intuitionistic logic. He called this system LJ. In LK any number of formulas is allowed to appear on the conclusion side of a sequent; in contrast LJ allows at most one formula in this position. Other derivatives of LK are limited to intuitionstic derivations but still allow multiple conclusions in a sequent. LJ' [l] is one example. Hilbert-style calculus Intuitionistic logic can be defined using the following Hilbert-style calculus. Compare with the deduction system at Propositional calculus# Alternative calculus. In propositional logic, the inference rule is modus ponens • MP: from (j) and (f> — > ip infer ip and the axioms are THEN-1: (j) _► ( x _> 0) THEN-2: (0 _> ( % _> ^)) AND-1: ^ Ax -> <f> AND-2: (/)A X ^ X ((^->x)->(0-^)) Intuitionistic logic • AND-3:0^( X ^(0A X )) • OR-1: — >(j) \/ x . OR-2: x ^(j)Vx . O R-3:(0-.V)-^((x->^)->(0Vx-^^)) • FALSE: _L — > To make this a system of first-order predicate logic, the generalization rules • V -GEN: from -0 — > (f> infer ^ — y (\/x 0) , if X is not free in ip • 3 -GEN: from — > ip infer (3:r 0) — > , if X is not free in if) are added, along with the axioms • PRED-1: (\/x 4>(x)) — > 0(f), if the term f is free for substitution for the variable x in (i.e., if no occurrence of any variable in t becomes bound in 0(f)) • PRED-2: 0(f) — ► CBx (f)(x)), with the same restriction as for PRED-1 Optional connectives Negation If one wishes to include a connective — i for negation rather than consider it an abbreviation for — >■ _L , it is enough to add: • NOT-1': (0 _► J_) _> ^0 • NOT-2': -n0 _► (0 _> J_) There are a number of alternatives available if one wishes to omit the connective J_ (false). For example, one may replace the three axioms FALSE, NOT-1', and NOT-2' with the two axioms to NOT-1 are (0 _► -,*) _> ( x _> ^0)c Equivalence The connective <— > for equivalence may be treated as an abbreviation, with <-> ^ standing for (0 — *■ X) A (X — ^ 0) • Alternatively, one may add the axioms ' IFF-1: (0 <-> *) -► (^ -► X) • IFF - 2; (0 <-> x) ^ (x ^ 0) ' IFF-3: (0 -> *) -► ((* -> 0) -> (0 <-► X )) IFF-1 and IFF-2 can, if desired, be combined into a single axiom (0 <-> x) — ► ((0 — ► x) A (X — *" 0)) usm 8 conjunction. . NOT-1: (0-> X )- • NOT-2: _> (-,0 - > ((0 -+ -x) -► ^0) as at Propositional calculus# Axioms. Alternatives (0 -> -.0) -► -.0 . Intuitionistic logic 10 Relation to classical logic The system of classical logic is obtained by adding any one of the following axioms: • V — >0 (Law of the excluded middle. May also be formulated as (0 — ► -)A — s- ((— .0 — > %) — y y).) • —1—10 — > (Double negation elimination) • ((0 — s- x) —> 0) —> <t> ( Peirce ' s law ) In general, one may take as the extra axiom any classical tautology that is not valid in the two-element Kripke frame o >o (in other words, that is not included in Smetanich's logic). Another relationship is given by the Godel— Gentzen negative translation, which provides an embedding of classical first-order logic into intuitionistic logic: a first-order formula is provable in classical logic if and only if its Godel— Gentzen translation is provable intuitionistically. Therefore intuitionistic logic can instead be seen as a means of extending classical logic with constructive semantics. Non-interdefinability of operators In classical propositional logic, it is possible to take one of conjunction, disjunction, or implication as primitive, and define the other two in terms of it together with negation, such as in Lukasiewicz's three axioms of propositional logic. It is even possible to define all four in terms of a sole sufficient operator such as the Peirce arrow (NOR) or Sheffer stroke (NAND). Similarly, in classical first-order logic, one of the quantifiers can be defined in terms of the other and negation. These are fundamentally consequences of the law of bivalence, which makes all such connectives merely Boolean functions. The law of bivalence does not hold in intuitionistic logic, only the law of non-contradiction. As a result none of the basic connectives can be dispensed with, and the above axioms are all necessary. Most of the classical identities are only theorems of intuitionistic logic in one direction, although some are theorems in both directions. They are as follows: Conjunction versus disjunction: • (0 A -0) -> ~>{-4 V -t0) • (0 V -0) -> ->{-4 A -t0) • (-10 V -i^) -> -.(0 A -0) • (-10 A -i^) <-> -.(0 V -0) Conjunction versus implication: . (0A*0)^-(0^^) • (0->V)->--(0A-^) • (0 A -i^) — » -i(0 — > tjj) • (0 — > ->ip) <-► -i(0 A tjj) Disjunction versus implication: • (0v?/o^H>^v) . (^0vv)^(0^V) • -h(0->*0) -► ^(^0V^0) • -i(0V^) <-► -i(i0 -> ^) Universal versus existential quantification: • (Vx 0(x)) -> -.(3x -0(x)) • (3x 0(x)) -> -.(Vx -^>{x)) • (3x -0(x)) -»■ -.(Vz 0(x)) • (Vx -0(x)) ^ -.(3a: 0(x)) Intuitionistic logic 1 1 So, for example, "a or b" is a stronger statement than "if not a, then b", whereas these are classically interchangeable. On the other hand, "neither a nor b" is equivalent to "not a, and also not b". If we include equivalence in the list of connectives, some of the connectives become definable from others: (^ <_, ^) <_, ((0 _>,£,) A ty _> 0)) (0 -> -0) •<-> ((0 V l/>) <-> </>) (0 -►,&) <->((<£ A V) «-><*) (0 A ^) <-> {((0 V -0) ^ ^) ^ 0) In particular, {v, <->, ±} and {v, <->, -i} are complete bases of intuitionistic connectives. As shown by Alexander Kuznetsov, either of the following defined connectives can serve the role of a sole sufficient operator for intuitionistic logic: • ((p V q) A -t) V (-ip A (g <-> r)), • p^(qA^rA(sVt)). Semantics The semantics are rather more complicated than for the classical case. A model theory can be given by Heyting algebras or, equivalently, by Kripke semantics. Heyting algebra semantics In classical logic, we often discuss the truth values that a formula can take. The values are usually chosen as the members of a Boolean algebra. The meet and join operations in the Boolean algebra are identified with the a and v logical connectives, so that the value of a formula of the form A a B is the meet of the value of A and the value of B in the Boolean algebra. Then we have the useful theorem that a formula is a valid sentence of classical logic if and only if its value is 1 for every valuation — that is, for any assignment of values to its variables. A corresponding theorem is true for intuitionistic logic, but instead of assigning each formula a value from a Boolean algebra, one uses values from a Heyting algebra, of which Boolean algebras are a special case. A formula is valid in intuitionistic logic if and only if it receives the value of the top element for any valuation on any Heyting algebra. It can be shown that to recognize valid formulas, it is sufficient to consider a single Heyting algebra whose elements are the open subsets of the real line R. In this algebra, the a and v operations correspond to set intersection and union, and the value assigned to a formula A — > B is int(A u B), the interior of the union of the value of B and the complement of the value of A. The bottom element is the empty set 0, and the top element is the entire line R. The negation -A of a formula A is (as usual) defined to be A — > 0. The value of -A then reduces to int(A ), the interior of the complement of the value of A, also known as the exterior of A. With these assignments, intuitionistically valid formulas are precisely those that are assigned the value of the entire line. For example, the formula ->(A a -A) is valid, because no matter what set X is chosen as the value of the formula A, the value of ->(A a -A) can be shown to be the entire line: Value(-.(A a -.A)) = int((Value(A a ^A)) C ) = int((Value(A) n Value(^A)) C ) = int((X n int((Value(A)) C )) C ) = int((X n intCX )) ) A theorem of topology tells us that int(A ) is a subset of a , so the intersection is empty, leaving: int(0 C ) = int(R) = R Intuitionistic logic 12 So the valuation of this formula is true, and indeed the formula is valid. But the law of the excluded middle, A v -A, can be shown to be invalid by letting the value of A be {y : y > }. Then the value of -A is the interior of {y : y < }, which is {y : y < }, and the value of the formula is the union of {y : y > } and {y : y < }, which is {y : y * }, not the entire line. The interpretation of any intuitionistically valid formula in the infinite Heyting algebra described above results in the top element, representing true, as the valuation of the formula, regardless of what values from the algebra are T31 assigned to the variables of the formula. Conversely, for every invalid formula, there is an assignment of values to the variables that yields a valuation that differs from the top element. No finite Heyting algebra has both these properties. Kripke semantics Building upon his work on semantics of modal logic, Saul Kripke created another semantics for intuitionistic logic, known as Kripke semantics or relational semantics. Relation to other logics Intutionistic logic is related by duality to a paraconsistent logic known as Brazilian, anti-intuitionistic or dual-intuitionistic logic. The subsystem of intuitionistic logic with the FALSE axiom removed is known as minimal logic. Notes [1] Proof Theory by G. Takeuti, ISBN0444 104925 [2] Alexander Chagrov, Michael Zakharyaschev, Modal Logic, vol. 35 of Oxford Logic Guides, Oxford University Press, 1997, pp. 58—59. [3] S0rensen, Morten Heine B; Pawel Urzyczyn (2006). Lectures on the Curry-Howard Isomorphism. Studies in Logic and the Foundations of Mathematics. Elsevier, p. 42. ISBN 0444520775. [4] Alfred Tarski, Der Aussagenkalkul und die Topologie, Fundamenta Mathematicae 31 (1938), 103—134. (http://matwbn.icm.edu.pl/tresc. php?wyd=l&tom=31) [5] Rasiowa, Helena; Roman Sikorski (1963). The Mathematics of Metamathematics. Monografie matematyczne. Warsaw: Pahstwowe Wydawn. Naukowe. pp. 385-386. [6] Intuitionistic Logic (http://plato.stanford.edu/entries/logic-intuitionistic/). Written by Joan Moschovakis (http://www.math.ucla.edu/ ~joan/). Published in Stanford Encyclopedia of Philosophy. [7] Aoyama, Hiroshi (2004). "LK, LJ, Dual Intuitionistic Logic, and Quantum Logic". Notre Dame Journal of Formal Logic 45 (4): 193—213. doi:10.1305/ndjfl/1099238445. References • Van Dalen, Dirk, 2001, "Intuitionistic Logic", in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell. • Morten H. S0rensen, Pawel Urzyczyn, 2006, Lectures on the Curry-Howard Isomorphism (chapter 2: "Intuitionistic Logic"). Studies in Logic and the Foundations of Mathematics vol. 149, Elsevier. • W. A. Carnielli (with A. B.M. Brunner). "Anti-intuitionism and paraconsistency" (http://dx.doi.Org/10.1016/j. jal.2004.07.016). Journal of Applied Logic Volume 3, Issue 1, March 2005, pages 161-184. Intuitionistic logic 13 External links • Stanford Encyclopedia of Philosophy: " Intuitionistic Logic (http://plato.stanford.edu/entries/ logic-intuitionistic/)" — by Joan Moschovakis. Heyting arithmetic In mathematical logic, Heyting arithmetic (sometimes abbreviated HA) is an axiomatization of arithmetic in accordance with the philosophy of intuitionism. It is named after Arend Heyting, who first proposed it. Heyting arithmetic adopts the axioms of Peano arithmetic (PA), but uses intuitionistic logic as its rules of inference. In particular, the law of the excluded middle does not hold in general, though the induction axiom can be used to prove many specific cases. For instance, one can prove that V x, yEN:x = yvx*yisa theorem (any two natural numbers are either equal to each other, or not equal to each other). In fact, since "=" is the only predicate symbol in Heyting arithmetic, it then follows that, for any quantifier-free formula p, V x, y, z, ■ ■ ■ G N : p v ->/? is a theorem (where x,y,z... are the free variables in/)). Kurt Godel studied the relationship between Heyting arithmetic and Peano arithmetic. He used the Godel— Gentzen negative translation to prove in 1933 that if HA is consistent, then PA is also consistent. Heyting arithmetic should not be confused with Heyting algebras, which are the intuitionistic analogue of Boolean algebras. External links • Stanford Encyclopedia of Philosophy: "Intuitionistic Number Theory by Joan Moschovakis. References [1] http://plato. Stanford. edu/entries/logic-intuitionistic/#IntNumTheHeyAri 14 Algebraic Logic and Many- Valued Logic Algebraic logic In mathematical logic, algebraic logic is the study of logic presented in an algebraic style. Algebras as models of logics Algebraic logic treats algebraic structures, often bounded lattices, as models (interpretations) of certain logics, making logic a branch of order theory. In algebraic logic: • Variables are tacitly universally quantified over some universe of discourse. There are no existentially quantified variables or open formulas; • Terms are built up from variables using primitive and defined operations. There are no connectives; • Formulas, built from terms in the usual way, can be equated if they are logically equivalent. To express a tautology, equate a formula with a truth value; • The rules of proof are the substitution of equals for equals, and uniform replacement. Modus ponens remains valid, but is seldom employed. In the table below, the left column contains one or more logical or mathematical systems, and the algebraic structure which are its models are shown on the right in the same row. Some of these structures are either Boolean algebras or proper extensions thereof. Modal and other nonclassical logics are typically modeled by what are called "Boolean algebras with operators." Algebraic formalisms going beyond first-order logic in at least some respects include: • Combinatory logic, having the expressive power of set theory; • Relation algebra, arguably the paradigmatic algebraic logic, can express Peano arithmetic and most axiomatic set theories, including the canonical ZFC. logical system its models Classical sentential logic Lindenbaum-Tarski algebra Two-element Boolean algebra Intuitionistic propositional logic Heyting algebra Lukasiewicz logic MV-algebra Modal logic K Modal algebra Lewis's S4 Interior algebra Lewis's S5; Monadic predicate logic Monadic Boolean algebra First-order logic Cylindric algebra Polyadic algebra Predicate functor logic Set theory Combinatory logic Relation algebra Algebraic logic 15 History On the history of algebraic logic before World War II, see Brady (2000) and Grattan-Guinness (2000) and their ample references. On the postwar history, see Maddux (1991) and Quine (1976). Algebraic logic has at least two meanings: • The study of Boolean algebra, begun by George Boole, and of relation algebra, begun by Augustus DeMorgan, extended by Charles Sanders Peirce, and taking definitive form in the work of Ernst Schroder; • Abstract algebraic logic, a branch of contemporary mathematical logic. Perhaps surprisingly, algebraic logic is the oldest approach to formal logic, arguably beginning with a number of memoranda Leibniz wrote in the 1680s, some of which were published in the 19th century and translated into English by Clarence Lewis in 1918. But nearly all of Leibniz's known work on algebraic logic was published only in 1903, after Louis Couturat discovered it in Leibniz's Nachlass. Parkinson (1966) and Loemker (1969) translated selections from Couturat's volume into English. Brady (2000) discusses the rich historical connections between algebraic logic and model theory. The founders of model theory, Ernst Schroder and Leopold Loewenheim, were logicians in the algebraic tradition. Alfred Tarski, the founder of set theoretic model theory as a major branch of contemporary mathematical logic, also: • Co-discovered Lindenbaum-Tarski algebra; • Invented cylindric algebra; • Wrote the 1941 paper that revived relation algebra, and that can be seen as the starting point of abstract algebraic logic. Modern mathematical logic began in 1847, with two pamphlets whose respective authors were Augustus DeMorgan and George Boole. They, and later C.S. Peirce, Hugh MacColl, Frege, Peano, Bertrand Russell, and A. N. Whitehead all shared Leibniz's dream of combining symbolic logic, mathematics, and philosophy. Relation algebra is arguably the culmination of Leibniz's approach to logic. With the exception of some writings by Leopold Loewenheim and Thoralf Skolem, algebraic logic went into eclipse soon after the 1910-13 publication of Principia Mathematica, not to revive until Tarski's 1940 reexposition of relation algebra. Leibniz had no influence on the rise of algebraic logic because his logical writings were little studied before the Parkinson and Loemker translations. Our present understanding of Leibniz the logician stems mainly from the work of Wolfgang Lenzen, summarized in Lenzen (2004). To see how present-day work in logic and metaphysics can draw inspiration from, and shed light on, Leibniz's thought, see Zalta (2000). References • Brady, Geraldine, 2000. From Peirce to Skolem: A neglected chapter in the history of logic. North-Holland/Elsevier Science BV: catalog page , Amsterdam, Netherlands, 625 pages. mi • Burris, Stanley, 2009. The Algebra of Logic Tradition . Stanford Encyclopedia of Philosophy. • Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots. Princeton Univ. Press. • Lenzen, Wolfgang, 2004, "Leibniz's Logic in Gabbay, D., and Woods, J., eds., Handbook of the History of Logic, Vol. 3: The Rise of Modern Logic from Leibniz to Frege. North-Holland: 1-84. • Loemker, Leroy (1969 (1956)), Leibniz: Philosophical Papers and Letters, Reidel. • Roger Maddux, 1991, "The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations," Studia Logica 50: 421-55. • Parkinson, G.H.R., 1966. Leibniz: Logical Papers. Oxford Uni. Press. • Willard Quine, 1976, "Algebraic Logic and Predicate Functors" in The Ways of Paradox. Harvard Univ. Press: 283-307. • Zalta, E. N., 2000, "A (Leibnizian) Theory of Concepts ," Philosophiegeschichte und logische Analyse / Logical Analysis and History of Philosophy 3: 137-183. Algebraic logic 16 External links • Stanford Encyclopedia of Philosophy: "Propositional Consequence Relations and Algebraic Logic — by Ramon Jansana. References [1] http://www.philosophie.uni-osnabrueck.de/Puhlikationen%20Lenzen/Lenzen%20Leibniz%20Logic.pdf [2] http://mally. stanford.edu/Papers/leibniz.pdf [3] http://www.elsevier.com/wps/find/bookdescription.cws_home/621535/description [4] http://plato.stanford.edu/entries/algebra-logic-tradition/ [5] http://mally.stanford.edu/leibniz.pdf [6] http://plato.stanford.edu/entries/consequence-algebraic/ Lukasiewicz logic In mathematics, Lukasiewicz logic (English pronunciation: /lulke'jevltj/, Polish pronunciation: [wuka'eev'itg]) is a non-classical, many valued logic. It was originally defined in the early 20th-century by Jan Lukasiewicz as a three-valued logic; it was later generalized to n-valued (for all finite n) as well as infinitely-many-valued variants, both propositional and first-order. It belongs to the classes of t-norm fuzzy logics and substructural logics. Language The propositional connectives of Lukasiewicz logic are implication — > , negation — i , equivalence <— > , weak conjunction A , strong conjunction (§) , weak disjunction V , strong disjunction © , and propositional constants Qand J. The presence of weak and strong conjunction and disjunction is a common feature of substructural logics without the rule of contraction, among which Lukasiewicz logic belongs. Axioms The original system of axioms for propositional infinite-valued Lukasiewicz logic used implication and negation as the primitive connectives: A -> (B -> A) (A^B)^((B^C)^(A^C)) ((A^B)^B)^((B^A)^A) {-^B^^A)^(A^B). Propositional infinite-valued Lukasiewicz logic can also be axiomatized by adding the following axioms to the axiomatic system of monoidal t-norm logic: • Divisibility: [A A B) -> (A (_) (A -> B)) • Double negation: — i— 1__ — $• __. That is, infinite-valued Lukasiewicz logic arises by adding the axiom of double negation to basic t-norm logic BL, or by adding the axiom of divisibility to the logic IMTL. Lukasiewicz logic 17 Real-valued semantics Infinite-valued Lukasiewicz logic is a real-valued logic in which sentences from sentential calculus may be assigned a truth value of not only zero or one but also any real number in between (eg. 0.25). Valuations have a recursive definition where: • w{6 o (/)) — F (w(9), w((j))yov a binary connective °, • w^e) = F^(w{0)), • w(0) = Oand w (l) = 1, and where the definitions of the operations hold as follows: • Implication: F^.(x, y) — min{l, 1 — x + y} • Equivalence: F„(x, y) — 1 — \x — y\ • Negation: F-,(x) — 1 — x • Weak Conjunction: F A (x, y) — min{z:, y} • Weak Disjunction: F v (x,y) — max{:r,y} • Strong Conjunction: F®(x, y) — max{0, x + y — 1} • Strong Disjunction: F @ (x, y) — min{l, x + y}. The truth function F^of strong conjunction is the Lukasiewicz t-norm and the truth function i^gof strong disjunction is its dual t-conorm. The truth function i* 1 is the residuum of the Lukasiewicz t-norm. All truth functions of the basic connectives are continuous. By definition, a formula is a tautology of infinite-valued Lukasiewicz logic if it evaluates to 1 under any valuation of propositional variables by real numbers in the interval [0, 1]. General algebraic semantics The standard real-valued semantics determined by the Lukasiewicz t-norm is not the only possible semantics of Lukasiewicz logic. General algebraic semantics of propositional infinite-valued Lukasiewicz logic is formed by the class of all MV-algebras. The standard real-valued semantics is a special MV-algebra, called the standard MV-algebra. Like other t-norm fuzzy logics, propositional infinite-valued Lukasiewicz logic enjoys completeness with respect to the class of all algebras for which the logic is sound (that is, MV-algebras) as well as with respect to only linear ones. This is expressed by the general, linear, and standard completeness theorems: The following conditions are equivalent: • A i s provable in propositional infinite-valued Lukasiewicz logic • A i s valid in all MV-algebras (general completeness) • J^ is valid in all linearly ordered MV-algebras {linear completeness) • J^ i s valid in the standard MV-algebra {standard completeness). References [1] Lukasiewicz J., 1920, O logice trojwartosciowej (in Polish). Ruch filozoficzny 5:170—171. English translation: On three-valued logic, in L. Borkowski (ed.), Selected works by Jan Lukasiewicz, North-Holland, Amsterdam, 1970, pp. 87-88. ISBN 0720422523 [2] Hay, L.S., 1963, Axiomatization of the infinite-valued predicate calculus. Journal of Symbolic Logic 28:77—86. [3] Hajek P., 1998, Metamathematics of Fuzzy Logic. Dordrecht: Kluwer. [4] Ono, H., 2003, "Substructural logics and residuated lattices — an introduction". In F.V. Hendricks, J. Malinowski (eds.): Trends in Logic: 50 Years of Studia Logica, Trends in Logic 20: 177-212. Ternary logic 18 Ternary logic A ternary, three-valued or trivalent logic (sometimes abbreviated 3VL) is any of several multi-valued logic systems in which there are three truth values indicating true, false and some indeterminate third value. This is contrasted with the more commonly known bivalent logics (such as classical sentential or boolean logics) which provide only for true and false. Conceptual form and basic ideas were initially created by Lukasiewicz, Lewis and Sulski. These were then re-formulated by Grigore Moisil in an axiomatic algebraic form, and also extended to « -valued logics in 1945. Definitions Concerning fuzziness, ternary logic might be seen formally as a fuzzy type of logic as a value may be different from just false (0) or true (1); however, ternary logic is defined as a crisp logic. Representation of values As with bivalent logic, truth values in ternary logic may be represented numerically using various representations of the ternary numeral system. A few of the more common examples are: • 1 for true, 2 for false, and for unknown, irrelevant, or both [1] [21 • fox false, 1 for true, and a third non-integer symbol such as # or Vi for the final value • Balanced ternary uses -1 for false, +1 for true and for the third value; these values may also be simplified to -, +, and 0, respectively. This article mainly illustrates a system of ternary propositional logic using the truth values {false, unknown, and true}, and extends conventional boolean connectives to a trivalent context. Ternary predicate logics exist as well; these may have readings of the quantifier different from classical (binary) predicate logic, and may include alternative quantifiers as well. Basic truth table Below is a truth table showing the logic operations for Kleene's logic. A B A ORB A ANDB NOT A True True True True False True Unknown True Unknown False True False True False False Unknown True True Unknown Unknown Unknown Unknown Unknown Unknown Unknown Unknown False Unknown False Unknown False True True False True False Unknown Unknown False True False False False False True In this truth table, the UNKNOWN state can be metaphorically thought of as a sealed box containing either an unambiguously TRUE or unambiguously FALSE value. The knowledge of whether any particular UNKNOWN state secretly represents TRUE or FALSE at any moment in time is not available. However, certain logical operations can yield an unambiguous result, even if they involve at least one UNKNOWN operand. For example, since TRUE OR TRUE equals TRUE, and TRUE OR FALSE also equals TRUE, one can infer that TRUE OR UNKNOWN equals Ternary logic 19 TRUE, as well. In this example, since either bivalent state could be underlying the UNKNOWN state, but either state also yields the same result, a definitive TRUE results in all three cases. In database applications The database structural query language SQL implements ternary logic as a means of handling NULL field content. SQL uses NULL to represent missing data in a database. If a field contains no defined value, SQL assumes this means that an actual value exists, but that value is not currently recorded in the database. Note that a missing value is not the same as either a numeric value of zero, or a string value of zero length. Comparing anything to NULL — even another NULL — results in an UNKNOWN truth state. For example, the SQL expression "City = 'Paris'" resolves to FALSE for a record with "Chicago" in the City field, but it resolves to UNKNOWN for a record with a NULL City field. In other words, to SQL, an undefined field represents potentially any possible value: a missing city might or might not represent Paris. Using ternary logic, SQL can then account for the UNKNOWN truth state in evaluating boolean expressions. Consider the expression "City = 'Paris' OR Balance < 0.0". This expression resolves to TRUE for any record whose Balance field contains a negative number. Likewise, this expression is TRUE for any record with 'Paris' in its City field. The expression resolves to FALSE only for a record whose City field explicitly contains a string other than 'Paris', and whose Balance field explicitly contains a non-negative number. In any other case, the expression resolves to UNKNOWN. This is because a missing City value might be missing the string 'Paris', and a missing Balance might be missing a negative number. However, regardless of missing data, a boolean OR operation is FALSE only when both of its operands are also FALSE, so not all missing data leads to an UNKNOWN resolution. In SQL Data Manipulation Language, a truth state of TRUE for an expression (e.g., in a WHERE clause) initiates an action on a row (e.g. return the row), while a truth state of UNKNOWN or FALSE does not. In this way, ternary logic is implemented in SQL, while behaving as binary logic to the SQL user. SQL Check Constraints behave differently, however. Only a truth state of FALSE results in a violation of a check constraint. A truth state of TRUE or UNKNOWN indicates a row has been successfully validated against the check constraint^ . Electronics Digital electronics theory supports four distinct logic values (as defined in VHDL's std_Iogic): 1 or High, usually representing TRUE. or Low, usually representing FALSE. X representing a "Conflict". U representing "Unassigned" or "Unknown". - representing "Don't Care". Z representing "high impedance", undriven line. H, L and W are other high-impedance values, the weak pull to "High", "Low" and "Don't Know" correspondingly. The "X" value does not exist in real-world circuits, it is merely a placeholder used in simulators and for design purposes. Some simulators support representation of the "Z" value, others do not. The "Z" value does exist in real-world circuits but only as an output state. Ternary logic 20 Use of "X" value in simulation Many hardware description language (HDL) simulation tools, such as Verilog and VHDL, support an unknown value like that shown above during simulation of digital electronics. The unknown value may be the result of a design error, which the designer can correct before synthesis into an actual circuit. The unknown also represents uninitialised memory values and circuit inputs before the simulation has asserted what the real input value should be. HDL synthesis tools usually produce circuits that operate only on binary logic. Use of "X" value in digital design When designing a digital circuit, some conditions may be outside the scope of the purpose that the circuit will perform. Thus, the designer does not care what happens under those conditions. In addition, the situation occurs that inputs to a circuit are masked by other signals so the value of that input has no effect on circuit behaviour. In these situations, it is traditional to use "X" as a placeholder to indicate "Don't Care" when building truth tables. This is especially common in state machine design and Karnaugh map simplification. The "X" values provide additional degrees of freedom to the final circuit design, generally resulting in a simplified and smaller circuit. Once the circuit design is complete and a real circuit is constructed, the "X" values will no longer exist. They will become some tangible "0" or " 1 " value but could be either depending on the final design optimisation. Use of "Z" value for high impedance Some digital devices support a form of three-state logic on their outputs only. The three states are "0", "1", and "Z". Commonly referred to as tristate logic (a trademark of National Semiconductor), it comprises the usual true and false states, with a third transparent high impedance state (or 'off-state') which effectively disconnects the logic output. This provides an effective way to connect several logic outputs to a single input, where all but one are put into the high impedance state, allowing the remaining output to operate in the normal binary sense. This is commonly used to connect banks of computer memory and other similar devices to a common data bus; a large number of devices can communicate over the same channel simply by ensuring only one is enabled at a time. It is important to note that while outputs can have one of three states, inputs can only recognise two. Hence the kind of relations shown in the table above do not occur. Although it could be argued that the high-impedance state is effectively an "unknown", there is absolutely no provision in the vast majority of normal electronics to interpret a high-impedance state as a state in itself. Inputs can only detect "0" and "1". When a digital input is left disconnected (i.e., when it is given a high impedance signal), the digital value interpreted by the input depends on the type of technology used. TTL technology will reliably default to a "1" state. On the other hand CMOS technology will temporarily hold the previous state seen on that input (due to the capacitance of the gate input). Over time, leakage current causes the CMOS input to drift in a random direction, possibly causing the input state to flip. Disconnected inputs on CMOS devices can pick up noise, they can cause oscillation, the supply current may dramatically increase (crowbar power) or the device may completely destroy itself. Exotic ternary-logic devices True ternary logic can be implemented in electronics, although the complexity of design has thus far made it uneconomical to pursue commercially and interest has been primarily confined to research, since 'normal' binary logic is much cheaper to implement and in most cases can easily be configured to emulate ternary systems. However, there are useful applications in fuzzy logic and error correction, and several true ternary logic devices have been manufactured (see external links). Ternary logic 2 1 References [1] Hayes, Brian (November-December, 2001). "Third Base". American Scientist (Sigma Xi, the Scientific Research Society) 89 (6): 490—494. doi: 10.151 1/2001 .6.490. [2] The Penguin Dictionary of Mathematics. 2nd Edition. London, England: Penguin Books. 1998. pp. 417. [3] Knuth, Donald E. (1981). The Art of Computer Programming Vol. 2. Reading, Mass.: Addison- Wesley Publishing Company, pp. 190. [4] Lex de Haan and Gennick, Jonathan (July-August, 2005). "Nulls: Nothing to Worry About" (http://www.oracle.com/technetwork/ issue-archive/2005/05-jul/o45sql-097727.html). Oracle Magazine (Oracle). . [5] Coles, Michael (February 26, 2007). "Null Versus Null?" (http://www.sqlservercentral.com/columnists/mcoles/2829.asp). SQL Server Central (Red Gate Software). . [6] Wakerly, John F (2001). Digital Design Principles & Practices. Prentice Hall. ISBN 0-13-090772-3. [7] National Semiconductor (1993), LS TTL Data Book (http://www.national.com), National Semiconductor Corporation, External links • Jeffs Trinary Wiki (archived) (http://web.archive.Org/web/20080525122206/http://jeff.tk/wiki/Trinary) • Steve Grubb's Trinary Website (archived) (http://web.archive.org/web/2008061 1055612/http://www. trinary. cc/) • Boost. Tribool (http://www.boost.org/doc/html/tribool.html) — an implementation of ternary logic in C++ • Team-R2D2 (http://www.inria.fr/rapportsactivite/RA2004/r2d22004/uid51.html) - a French institute that fabricated the first full-ternary logic chip (a 64-tert SRAM and 4-tert adder) in 2004 • A polar place value number system for computers and life in general (http://www.abhijit.info/tristate/tristate. html) • Applet on Ternary Char Representation (http://tvm.manojky.net) Multi-valued logic Multi-valued logics are 'logical calculi' in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false") for any proposition. An obvious extension to classical two-valued logic is an n-valued logic for n > 2. Those most popular in the literature are three-valued (e.g., Lukasiewicz's and Kleene's), — which accept the values "true", "false", and "unknown", — the finite-valued with more than 3 values, and the infinite-valued (e.g. fuzzy logic) logics. Relation to classical logic Logics are usually systems intended to codify rules for preserving some semantic property of propositions across transformations. In classical logic, this property is "truth." In a valid argument, the truth of the derived proposition is guaranteed if the premises are jointly true, because the application of valid steps preserves the property. However, that property doesn't have to be that of "truth"; instead, it can be some other concept. Multi-valued logics are intended to preserve the property of designationhood (or being designated). Since there are more than two truth values, rules of inference may be intended to preserve more than just whichever corresponds (in the relevant sense) to truth. For example, in a three-valued logic, sometimes the two greatest truth-values (when they are represented as e.g. positive integers) are designated and the rules of inference preserve these values. Precisely, a valid argument will be such that the value of the premises taken jointly will always be less than or equal to the conclusion. For example, the preserved property could be justification, the foundational concept of intuitionistic logic. Thus, a proposition is not true or false; instead, it is justified or flawed. A key difference between justification and truth, in this case, is that the law of excluded middle doesn't hold: a proposition that is not flawed is not necessarily justified; instead, it's only not proven that it's flawed. The key difference is the determinacy of the preserved property: One Multi-valued logic 22 may prove that P is justified, that P is flawed, or be unable to prove either. A valid argument preserves justification across transformations, so a proposition derived from justified propositions is still justified. However, there are proofs in classical logic that depend upon the law of excluded middle; since that law is not usable under this scheme, there are propositions that cannot be proven that way. Relation to fuzzy logic Multi-valued logic is strictly related with fuzzy set theory and fuzzy logic. The notion of fuzzy subset was introduced by Lotfi Zadeh as a formalization of vagueness; i.e., the phenomenon that a predicate may apply to an object not absolutely, but to a certain degree, and that there may be borderline cases. Indeed, as in multi-valued logic, fuzzy logic admits truth values different from "true" and "false". As an example, usually the set of possible truth values is the whole interval [0,1]. Nevertheless, the main difference between fuzzy logic and multi-valued logic is in the aims. In fact, in spite of its philosophical interest (it can be used to deal with the Sorites paradox), fuzzy logic is devoted mainly to the applications. More precisely, there are two approaches to fuzzy logic. The first one is very closely linked with multi-valued logic tradition (Hajek school). So a set of designed values is fixed and this enables us to define an entailment relation. The deduction apparatus is defined by a suitable set of logical axioms and suitable inference rules. Another approach (Goguen, Pavelka and others) is devoted to defining a deduction apparatus in which approximate reasonings are admitted. Such an apparatus is defined by a suitable fuzzy subset of logical axioms and by a suitable set of fuzzy inference rules. In the first case the logical consequence operator gives the set of logical consequence of a given set of axioms. In the latter the logical consequence operator gives the fuzzy subset of logical consequence of a given fuzzy subset of hypotheses. Another example of an infinitely-valued logic is probability logic. History The first known classical logician who didn't fully accept the law of excluded middle was Aristotle (who, ironically, is also generally considered to be the first classical logician and the "father of logic" ), who admitted that his laws did not all apply to future events (De Interpretatione, ch. IX). But he didn't create a system of multi-valued logic to explain this isolated remark. The later logicians until the coming of the 20th century followed Aristotelian logic, which includes or assumes the law of the excluded middle. The 20th century brought the idea of multi-valued logic back. The Polish logician and philosopher Jan Lukasiewicz began to create systems of many-valued logic in 1920, using a third value "possible" to deal with Aristotle's paradox of the sea battle. Meanwhile, the American mathematician Emil L. Post (1921) also introduced the formulation of additional truth degrees with n > 2, where n are the truth values. Later Jan Lukasiewicz and Alfred Tarski together formulated a logic on n truth values where n > 2 and in 1932 Hans Reichenbach formulated a logic of many truth values where «— >infinity. Kurt Godel in 1932 showed that intuitionistic logic is not a finitely-many valued logic, and defined a system of Godel logics intermediate between classical and intuitionistic logic; such logics are known as intermediate logics. Multi-valued logic 23 References • Beziau J.-Y. 1997 What is many-valued logic ? Proceedings of the 27th International Symposium on Multiple-Valued Logic, IEEE Computer Society, Los Alamitos, pp. 117—121. • Chang C.C. and Keisler H. J. 1966. Continuous Model Theory, Princeton, Princeton University Press. • Cignoli, R. L. O., D'Ottaviano, I, M. L., Mundici, D., 2000. Algebraic Foundations of Many-valued Reasoning. Kluwer. • Hajek P., 1998, Metamathematics of fuzzy logic. Kluwer. • Malinowski, Gregorz, 2001, Many-Valued Logics, in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell. • Gerla G. 2001, Fuzzy logic: Mathematical Tools for Approximate Reasoning, Kluwer Academic Publishers, Dordrecht. • Goguen J. A. 1968/69, The logic of inexact concepts, Synthese, 19, 325-373. • S. Gottwald, A Treatise on Many-Valued Logics. Studies in Logic and Computation, vol. 9, Research Studies Press: Baldock, Hertfordshire, England, 2001. • Pavelka J. 1979, On fuzzy logic I: Many-valued rules of inference, Zeitschr. f. math. Logik und Grundlagen d. Math., 25, 45-52. • Prior A. 1957, Time and Modality. Oxford University Press, based on his 1956 John Locke lectures External links T21 • Stanford Encyclopedia of Philosophy: "Many-Valued Logic — by Siegfried Gottwald. Notes [1] Hurley, Patrick. A Concise Introduction to Logic, 9th edition. (2006). [2] http://plato.stanford.edu/entries/logic-manyvalued/ Mathematical logic 24 Mathematical logic Mathematical logic (also known as symbolic logic) is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order logic, and definability. In computer science (particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in this article; see logic in computer science for those. Since its inception, mathematical logic has contributed to, and has been motivated by, the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Godel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems, rather than trying to find theories in which all of mathematics can be developed. History Mathematical logic emerged in the mid- 19th century as a subfield of mathematics independent of the traditional study of logic (Ferreiros 2001, p. 443). Before this emergence, logic was studied with rhetoric, through the syllogism, and with philosophy. The first half of the 20th century saw an explosion of fundamental results, accompanied by vigorous debate over the foundations of mathematics. Early history Sophisticated theories of logic were developed in many cultures, including China, India, Greece and the Islamic world. In the 18th century, attempts to treat the operations of formal logic in a symbolic or algebraic way had been made by philosophical mathematicians including Leibniz and Lambert, but their labors remained isolated and little known. 19th century In the middle of the nineteenth century, George Boole and then Augustus De Morgan presented systematic mathematical treatments of logic. Their work, building on work by algebraists such as George Peacock, extended the traditional Aristotelian doctrine of logic into a sufficient framework for the study of foundations of mathematics (Katz 1998, p. 686). Charles Sanders Peirce built upon the work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from 1870 to 1885. Gottlob Frege presented an independent development of logic with quantifiers in his Begriffsschrift, published in 1879, a work generally considered as marking a turning point in the history of logic. Frege's work remained obscure, however, until Bertrand Russell began to promote it near the turn of the century. The two-dimensional notation Frege developed was never widely adopted and is unused in contemporary texts. From 1890 to 1905, Ernst Schroder published Vorlesungen iiber die Algebra der Logik in three volumes. This work summarized and extended the work of Boole, De Morgan, and Peirce, and was a comprehensive reference to Mathematical logic 25 symbolic logic as it was understood at the end of the 19th century. Foundational theories Some concerns that mathematics had not been built on a proper foundation led to the development of axiomatic systems for fundamental areas of mathematics such as arithmetic, analysis, and geometry. In logic, the term arithmetic refers to the theory of the natural numbers. Giuseppe Peano (1888) published a set of axioms for arithmetic that came to bear his name (Peano axioms), using a variation of the logical system of Boole and Schroder but adding quantifiers. Peano was unaware of Frege's work at the time. Around the same time Richard Dedekind showed that the natural numbers are uniquely characterized by their induction properties. Dedekind (1888) proposed a different characterization, which lacked the formal logical character of Peano's axioms. Dedekind's work, however, proved theorems inaccessible in Peano's system, including the uniqueness of the set of natural numbers (up to isomorphism) and the recursive definitions of addition and multiplication from the successor function and mathematical induction. In the mid-19th century, flaws in Euclid's axioms for geometry became known (Katz 1998, p. 774). In addition to the independence of the parallel postulate, established by Nikolai Lobachevsky in 1826 (Lobachevsky 1840), mathematicians discovered that certain theorems taken for granted by Euclid were not in fact provable from his axioms. Among these is the theorem that a line contains at least two points, or that circles of the same radius whose centers are separated by that radius must intersect. Hilbert (1899) developed a complete set of axioms for geometry, building on previous work by Pasch (1882). The success in axiomatizing geometry motivated Hilbert to seek complete axiomatizations of other areas of mathematics, such as the natural numbers and the real line. This would prove to be a major area of research in the first half of the 20th century. The 19th century saw great advances in the theory of real analysis, including theories of convergence of functions and Fourier series. Mathematicians such as Karl Weierstrass began to construct functions that stretched intuition, such as nowhere-differentiable continuous functions. Previous conceptions of a function as a rule for computation, or a smooth graph, were no longer adequate. Weierstrass began to advocate the arithmetization of analysis, which sought to axiomatize analysis using properties of the natural numbers. The modern (e, 5)-definition of limit and continuous functions was already developed by Bolzano in 1817 (Felscher 2000), but remained relatively unknown. Cauchy in 1821 defined continuity in terms of infinitesimals (see Cours d'Analyse, page 34). In 1858, Dedekind proposed a definition of the real numbers in terms of Dedekind cuts of rational numbers (Dedekind 1872), a definition still employed in contemporary texts. Georg Cantor developed the fundamental concepts of infinite set theory. His early results developed the theory of cardinality and proved that the reals and the natural numbers have different cardinalities (Cantor 1874). Over the next twenty years, Cantor developed a theory of transfinite numbers in a series of publications. In 1891, he published a new proof of the uncountability of the real numbers that introduced the diagonal argument, and used this method to prove Cantor's theorem that no set can have the same cardinality as its powerset. Cantor believed that every set could be well-ordered, but was unable to produce a proof for this result, leaving it as an open problem in 1895 (Katz 1998, p. 807). Mathematical logic 26 20th century In the early decades of the 20th century, the main areas of study were set theory and formal logic. The discovery of paradoxes in informal set theory caused some to wonder whether mathematics itself is inconsistent, and to look for proofs of consistency. In 1900, Hilbert posed a famous list of 23 problems for the next century. The first two of these were to resolve the continuum hypothesis and prove the consistency of elementary arithmetic, respectively; the tenth was to produce a method that could decide whether a multivariate polynomial equation over the integers has a solution. Subsequent work to resolve these problems shaped the direction of mathematical logic, as did the effort to resolve Hilbert's Entscheidungsproblem, posed in 1928. This problem asked for a procedure that would decide, given a formalized mathematical statement, whether the statement is true or false. Set theory and paradoxes Ernst Zermelo (1904) gave a proof that every set could be well-ordered, a result Georg Cantor had been unable to obtain. To achieve the proof, Zermelo introduced the axiom of choice, which drew heated debate and research among mathematicians and the pioneers of set theory. The immediate criticism of the method led Zermelo to publish a second exposition of his result, directly addressing criticisms of his proof (Zermelo 1908a). This paper led to the general acceptance of the axiom of choice in the mathematics community. Skepticism about the axiom of choice was reinforced by recently discovered paradoxes in naive set theory. Cesare Burali-Forti (1897) was the first to state a paradox: the Burali-Forti paradox shows that the collection of all ordinal numbers cannot form a set. Very soon thereafter, Bertrand Russell discovered Russell's paradox in 1901, and Jules Richard (1905) discovered Richard's paradox. Zermelo (1908b) provided the first set of axioms for set theory. These axioms, together with the additional axiom of replacement proposed by Abraham Fraenkel, are now called Zermelo— Fraenkel set theory (ZF). Zermelo's axioms incorporated the principle of limitation of size to avoid Russell's paradox. In 1910, the first volume of Principia Mathematica by Russell and Alfred North Whitehead was published. This seminal work developed the theory of functions and cardinality in a completely formal framework of type theory, which Russell and Whitehead developed in an effort to avoid the paradoxes. Principia Mathematica is considered one of the most influential works of the 20th century, although the framework of type theory did not prove popular as a foundational theory for mathematics (Ferreiros 2001, p. 445). Fraenkel (1922) proved that the axiom of choice cannot be proved from the remaining axioms of Zermelo's set theory with urelements. Later work by Paul Cohen (1966) showed that the addition of urelements is not needed, and the axiom of choice is unprovable in ZF. Cohen's proof developed the method of forcing, which is now an important tool for establishing independence results in set theory. Symbolic logic Leopold Lowenheim (1915) and Thoralf Skolem (1920) obtained the Lowenheim— Skolem theorem, which says that first-order logic cannot control the cardinalities of infinite structures. Skolem realized that this theorem would apply to first-order formalizations of set theory, and that it implies any such formalization has a countable model. This counterintuitive fact became known as Skolem's paradox. In his doctoral thesis, Kurt Godel (1929) proved the completeness theorem, which establishes a correspondence between syntax and semantics in first-order logic. Godel used the completeness theorem to prove the compactness theorem, demonstrating the finitary nature of first-order logical consequence. These results helped establish first-order logic as the dominant logic used by mathematicians. In 1931, Godel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems, which proved the incompleteness (in a different meaning of the word) of all sufficiently strong, effective first-order theories. This result, known as Godel's incompleteness theorem, establishes severe limitations on axiomatic Mathematical logic 27 foundations for mathematics, striking a strong blow to Hilbert's program. It showed the impossibility of providing a consistency proof of arithmetic within any formal theory of arithmetic. Hilbert, however, did not acknowledge the importance of the incompleteness theorem for some time. Godel's theorem shows that a consistency proof of any sufficiently strong, effective axiom system cannot be obtained in the system itself, if the system is consistent, nor in any weaker system. This leaves open the possibility of consistency proofs that cannot be formalized within the system they consider. Gentzen (1936) proved the consistency of arithmetic using a finitistic system together with a principle of transfinite induction. Gentzen's result introduced the ideas of cut elimination and proof-theoretic ordinals, which became key tools in proof theory. Godel (1958) gave a different consistency proof, which reduces the consistency of classical arithmetic to that of intutitionistic arithmetic in higher types. Beginnings of the other branches Alfred Tarski developed the basics of model theory. Beginning in 1935, a group of prominent mathematicians collaborated under the pseudonym Nicolas Bourbaki to publish a series of encyclopedic mathematics texts. These texts, written in an austere and axiomatic style, emphasized rigorous presentation and set-theoretic foundations. Terminology coined by these texts, such as the words bijection, injection, and surjection, and the set-theoretic foundations the texts employed, were widely adopted throughout mathematics. The study of computability came to be known as recursion theory, because early formalizations by Godel and Kleene relied on recursive definitions of functions. When these definitions were shown equivalent to Turing's formalization involving Turing machines, it became clear that a new concept — the computable function — had been discovered, and that this definition was robust enough to admit numerous independent characterizations. In his work on the incompleteness theorems in 1931, Godel lacked a rigorous concept of an effective formal system; he immediately realized that the new definitions of computability could be used for this purpose, allowing him to state the incompleteness theorems in generality that could only be implied in the original paper. Numerous results in recursion theory were obtained in the 1940s by Stephen Cole Kleene and Emil Leon Post. Kleene (1943) introduced the concepts of relative computability, foreshadowed by Turing (1939), and the arithmetical hierarchy. Kleene later generalized recursion theory to higher-order functionals. Kleene and Kreisel studied formal versions of intuitionistic mathematics, particularly in the context of proof theory. Subfields and scope The Handbook of Mathematical Logic makes a rough division of contemporary mathematical logic into four areas: 1 . set theory 2. model theory 3. recursion theory, and 4. proof theory and constructive mathematics (considered as parts of a single area). Each area has a distinct focus, although many techniques and results are shared between multiple areas. The border lines between these fields, and the lines between mathematical logic and other fields of mathematics, are not always sharp. Godel's incompleteness theorem marks not only a milestone in recursion theory and proof theory, but has also led to Lob's theorem in modal logic. The method of forcing is employed in set theory, model theory, and recursion theory, as well as in the study of intuitionistic mathematics. The mathematical field of category theory uses many formal axiomatic methods, and includes the study of categorical logic, but category theory is not ordinarily considered a subfield of mathematical logic. Because of its applicability in diverse fields of mathematics, mathematicians including Saunders Mac Lane have proposed category theory as a foundational system for mathematics, independent of set theory. These foundations use toposes, which Mathematical logic 28 resemble generalized models of set theory that may employ classical or nonclassical logic. Formal logical systems At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems. These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language, or signature. The system of first-order logic is the most widely studied today, because of its applicability to foundations of mathematics and because of its desirable proof-theoretic properties. Stronger classical logics such as second-order logic or infinitary logic are also studied, along with nonclassical logics such as intuitionistic logic. First-order logic First-order logic is a particular formal system of logic. Its syntax involves only finite expressions as well-formed formulas, while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse. Early results about formal logic established limitations of first-order logic. The Lowenheim— Skolem theorem (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality. This shows that it is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to isomorphism. As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark. Godel's completeness theorem (Godel 1929) established the equivalence between semantic and syntactic definitions of logical consequence in first-order logic. It shows that if a particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. The compactness theorem first appeared as a lemma in Godel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory, and they are a key reason for the prominence of first-order logic in mathematics. Godel's incompleteness theorems (Godel 1931) establish additional limits on first-order axiomatizations. The first incompleteness theorem states that for any sufficiently strong, effectively given logical system there exists a statement which is true but not provable within that system. Here a logical system is effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom. A logical system is sufficiently strong if it can express the Peano axioms. When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not elementarily equivalent, a stronger limitation than the one established by the Lowenheim— Skolem theorem. The second incompleteness theorem states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that Hilbert's program cannot be completed. Mathematical logic 29 Other classical logics Many logics besides first-order logic are studied. These include infinitary logics, which allow for formulas to provide an infinite amount of information, and higher-order logics, which include a portion of set theory directly in their semantics. The most well studied infinitary logic is L ljJ1 ^ . In this logic, quantifiers may only be nested to finite depths, as in first order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula of L LJl ^such as (x = 0) V (x = 1) V (z = 2) V ■ • ■ • Higher-order logics allow for quantification not only of elements of the domain of discourse, but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having a separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type. The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of the completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis. Another type of logics are fixed-point logics that allow inductive definitions, like one writes for primitive recursive functions. One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or fuzzy logic. Lindstrom's theorem implies that the only extension of first-order logic satisfying both the Compactness theorem and the Downward Lowenheim— Skolem theorem is first-order logic. Nonclassical and modal logic Modal logics include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability (Solovay 1976) and set-theoretic forcing (Hamkins and Lowe 2007). Intuitionistic logic was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the law of the excluded middle, which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic is computable; this is not true in classical theories of arithmetic such as Peano arithmetic. Algebraic logic Algebraic logic uses the methods of abstract algebra to study the semantics of formal logics. A fundamental example is the use of Boolean algebras to represent truth values in classical propositional logic, and the use of Heyting algebras to represent truth values in intuitionistic propositional logic. Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such as cylindric algebras. Set theory Set theory is the study of sets, which are abstract collections of objects. Many of the basic notions, such as ordinal and cardinal numbers, were developed informally by Cantor before formal axiomatizations of set theory were developed. The first such axiomatization, due to Zermelo (1908b), was extended slightly to become Zermelo— Fraenkel set theory (ZF), which is now the most widely used foundational theory for mathematics. Mathematical logic 30 Other formalizations of set theory have been proposed, including von Neumann— Bernays— Godel set theory (NBG), Morse— Kelley set theory (MK), and New Foundations (NF). Of these, ZF, NBG, and MK are similar in describing a cumulative hierarchy of sets. New Foundations takes a different approach; it allows objects such as the set of all sets at the cost of restrictions on its set-existence axioms. The system of Kripke-Platek set theory is closely related to generalized recursion theory. Two famous statements in set theory are the axiom of choice and the continuum hypothesis. The axiom of choice, first stated by Zermelo (1904), was proved independent of ZF by Fraenkel (1922), but has come to be widely accepted by mathematicians. It states that given a collection of nonempty sets there is a single set C that contains exactly one element from each set in the collection. The set C is said to "choose" one element from each set in the collection. While the ability to make such a choice is considered obvious by some, since each set in the collection is nonempty, the lack of a general, concrete rule by which the choice can be made renders the axiom nonconstructive. Stefan Banach and Alfred Tarski (1924) showed that the axiom of choice can be used to decompose a solid ball into a finite number of pieces which can then be rearranged, with no scaling, to make two solid balls of the original size. This theorem, known as the Banach-Tarski paradox, is one of many counterintuitive results of the axiom of choice. The continuum hypothesis, first proposed as a conjecture by Cantor, was listed by David Hilbert as one of his 23 problems in 1900. Godel showed that the continuum hypothesis cannot be disproven from the axioms of Zermelo— Fraenkel set theory (with or without the axiom of choice), by developing the constructible universe of set theory in which the continuum hypothesis must hold. In 1963, Paul Cohen showed that the continuum hypothesis cannot be proven from the axioms of Zermelo— Fraenkel set theory (Cohen 1966). This independence result did not completely settle Hilbert's question, however, as it is possible that new axioms for set theory could resolve the hypothesis. Recent work along these lines has been conducted by W. Hugh Woodin, although its importance is not yet clear (Woodin 2001). Contemporary research in set theory includes the study of large cardinals and determinacy. Large cardinals are cardinal numbers with particular properties so strong that the existence of such cardinals cannot be proved in ZFC. The existence of the smallest large cardinal typically studied, an inaccessible cardinal, already implies the consistency of ZFC. Despite the fact that large cardinals have extremely high cardinality, their existence has many ramifications for the structure of the real line. Determinacy refers to the possible existence of winning strategies for certain two-player games (the games are said to be determined). The existence of these strategies implies structural properties of the real line and other Polish spaces. Model theory Model theory studies the models of various formal theories. Here a theory is a set of formulas in a particular formal logic and signature, while a model is a structure that gives a concrete interpretation of the theory. Model theory is closely related to universal algebra and algebraic geometry, although the methods of model theory focus more on logical considerations than those fields. The set of all models of a particular theory is called an elementary class; classical model theory seeks to determine the properties of models in a particular elementary class, or determine whether certain classes of structures form elementary classes. The method of quantifier elimination can be used to show that definable sets in particular theories cannot be too complicated. Tarski (1948) established quantifier elimination for real-closed fields, a result which also shows the theory of the field of real numbers is decidable. (He also noted that his methods were equally applicable to algebraically closed fields of arbitrary characteristic.) A modern subfield developing from this is concerned with o-minimal structures. Morley's categoricity theorem, proved by Michael D. Morley (1965), states that if a first-order theory in a countable language is categorical in some uncountable cardinality, i.e. all models of this cardinality are isomorphic, then it is categorical in all uncountable cardinalities. Mathematical logic 3 1 A trivial consequence of the continuum hypothesis is that a complete theory with less than continuum many nonisomorphic countable models can have only countably many. Vaught's conjecture, named after Robert Lawson Vaught, says that this is true even independently of the continuum hypothesis. Many special cases of this conjecture have been established. Recursion theory Recursion theory, also called computability theory, studies the properties of computable functions and the Turing degrees, which divide the uncomputable functions into sets which have the same level of uncomputability. Recursion theory also includes the study of generalized computability and definability. Recursion theory grew from of the work of Alonzo Church and Alan Turing in the 1930s, which was greatly extended by Kleene and Post in the 1940s. Classical recursion theory focuses on the computability of functions from the natural numbers to the natural numbers. The fundamental results establish a robust, canonical class of computable functions with numerous independent, equivalent characterizations using Turing machines, X calculus, and other systems. More advanced results concern the structure of the Turing degrees and the lattice of recursively enumerable sets. Generalized recursion theory extends the ideas of recursion theory to computations that are no longer necessarily finite. It includes the study of computability in higher types as well as areas such as hyperarithmetical theory and a-recursion theory. Contemporary research in recursion theory includes the study of applications such as algorithmic randomness, computable model theory, and reverse mathematics, as well as new results in pure recursion theory. Algorithmically unsolvable problems An important subfield of recursion theory studies algorithmic unsolvability; a decision problem or function problem is algorithmically unsolvable if there is no possible computable algorithm which returns the correct answer for all legal inputs to the problem. The first results about unsolvability, obtained independently by Church and Turing in 1936, showed that the Entscheidungsproblem is algorithmically unsolvable. Turing proved this by establishing the unsolvability of the halting problem, a result with far-ranging implications in both recursion theory and computer science. There are many known examples of undecidable problems from ordinary mathematics. The word problem for groups was proved algorithmically unsolvable by Pyotr Novikov in 1955 and independently by W. Boone in 1959. The busy beaver problem, developed by Tibor Rado in 1962, is another well-known example. Hilbert's tenth problem asked for an algorithm to determine whether a multivariate polynomial equation with integer coefficients has a solution in the integers. Partial progress was made by Julia Robinson, Martin Davis and Hilary Putnam. The algorithmic unsolvability of the problem was proved by Yuri Matiyasevich in 1970 (Davis 1973). Proof theory and constructive mathematics Proof theory is the study of formal proofs in various logical deduction systems. These proofs are represented as formal mathematical objects, facilitating their analysis by mathematical techniques. Several deduction systems are commonly considered, including Hilbert-style deduction systems, systems of natural deduction, and the sequent calculus developed by Gentzen. The study of constructive mathematics, in the context of mathematical logic, includes the study of systems in non-classical logic such as intuitionistic logic, as well as the study of predicative systems. An early proponent of predicativism was Hermann Weyl, who showed it is possible to develop a large part of real analysis using only predicative methods (Weyl 1918). Because proofs are entirely finitary, whereas truth in a structure is not, it is common for work in constructive mathematics to emphasize provability. The relationship between provability in classical (or nonconstructive) systems Mathematical logic 32 and provability in intuitionistic (or constructive, respectively) systems is of particular interest. Results such as the Godel— Gentzen negative translation show that it is possible to embed (or translate) classical logic into intuitionistic logic, allowing some properties about intuitionistic proofs to be transferred back to classical proofs. Recent developments in proof theory include the study of proof mining by Ulrich Kohlenbach and the study of proof- theoretic ordinals by Michael Rathjen. Connections with computer science The study of computability theory in computer science is closely related to the study of computability in mathematical logic. There is a difference of emphasis, however. Computer scientists often focus on concrete programming languages and feasible computability, while researchers in mathematical logic often focus on computability as a theoretical concept and on noncomputability. The theory of semantics of programming languages is related to model theory, as is program verification (in particular, model checking). The Curry— Howard isomorphism between proofs and programs relates to proof theory, especially intuitionistic logic. Formal calculi such as the lambda calculus and combinatory logic are now studied as idealized programming languages. Computer science also contributes to mathematics by developing techniques for the automatic checking or even finding of proofs, such as automated theorem proving and logic programming. Descriptive complexity theory relates logics to computational complexity. The first significant result in this area, Fagin's theorem (1974) established that NP is precisely the set of languages expressible by sentences of existential second-order logic. Foundations of mathematics In the 19th century, mathematicians became aware of logical gaps and inconsistencies in their field. It was shown that Euclid's axioms for geometry, which had been taught for centuries as an example of the axiomatic method, were incomplete. The use of infinitesimals, and the very definition of function, came into question in analysis, as pathological examples such as Weierstrass' nowhere-differentiable continuous function were discovered. Cantor's study of arbitrary infinite sets also drew criticism. Leopold Kronecker famously stated "God made the integers; all else is the work of man," endorsing a return to the study of finite, concrete objects in mathematics. Although Kronecker's argument was carried forward by constructivists in the 20th century, the mathematical community as a whole rejected them. David Hilbert argued in favor of the study of the infinite, saying "No one shall expel us from the Paradise that Cantor has created." Mathematicians began to search for axiom systems that could be used to formalize large parts of mathematics. In addition to removing ambiguity from previously-naive terms such as function, it was hoped that this axiomatization would allow for consistency proofs. In the 19th century, the main method of proving the consistency of a set of axioms was to provide a model for it. Thus, for example, non-Euclidean geometry can be proved consistent by defining point to mean a point on a fixed sphere and line to mean a great circle on the sphere. The resulting structure, a model of elliptic geometry, satisfies the axioms of plane geometry except the parallel postulate. With the development of formal logic, Hilbert asked whether it would be possible to prove that an axiom system is consistent by analyzing the structure of possible proofs in the system, and showing through this analysis that it is impossible to prove a contradiction. This idea led to the study of proof theory. Moreover, Hilbert proposed that the analysis should be entirely concrete, using the term finitary to refer to the methods he would allow but not precisely defining them. This project, known as Hilbert's program, was seriously affected by Godel's incompleteness theorems, which show that the consistency of formal theories of arithmetic cannot be established using methods formalizable in those theories. Gentzen showed that it is possible to produce a proof of the consistency of arithmetic in a finitary system augmented with axioms of transfinite induction, and the techniques he developed to so do were seminal in Mathematical logic 33 proof theory. A second thread in the history of foundations of mathematics involves nonclassical logics and constructive mathematics. The study of constructive mathematics includes many different programs with various definitions of constructive. At the most accommodating end, proofs in ZF set theory that do not use the axiom of choice are called constructive by many mathematicians. More limited versions of constructivism limit themselves to natural numbers, number-theoretic functions, and sets of natural numbers (which can be used to represent real numbers, facilitating the study of mathematical analysis). A common idea is that a concrete means of computing the values of the function must be known before the function itself can be said to exist. In the early 20th century, Luitzen Egbertus Jan Brouwer founded intuitionism as a philosophy of mathematics. This philosophy, poorly understood at first, stated that in order for a mathematical statement to be true to a mathematician, that person must be able to intuit the statement, to not only believe its truth but understand the reason for its truth. A consequence of this definition of truth was the rejection of the law of the excluded middle, for there are statements that, according to Brouwer, could not be claimed to be true while their negations also could not be claimed true. Brouwer's philosophy was influential, and the cause of bitter disputes among prominent mathematicians. Later, Kleene and Kreisel would study formalized versions of intuitionistic logic (Brouwer rejected formalization, and presented his work in unformalized natural language). With the advent of the BHK interpretation and Kripke models, intuitionism became easier to reconcile with classical mathematics. See also List of mathematical logic topics List of computability and complexity topics List of set theory topics List of first-order theories Knowledge representation Metalogic Logic symbols Notes [1] Undergraduate texts include Boolos, Burgess, and Jeffrey (2002), Enderton (2001), and Mendelson (1997). A classic graduate text by Shoenfield (2001) first appeared in 1967. [2] A detailed study of this terminology is given by Soare (1996). [3] Ferreiros (2001) surveys the rise of first-order logic over other formal logics in the early 20th century. References Undergraduate texts • ; Burgess, John; (2002), Computability and Logic (4th ed.), Cambridge: Cambridge University Press, ISBN 9780521007580 (pb.). • Enderton, Herbert (2001), A mathematical introduction to logic (2nd ed.), Boston, MA: Academic Press, ISBN 978-0-12-238452-3. • Hamilton, A.G. (1988), Logic for Mathematicians (2nd ed.), Cambridge: Cambridge University Press, ISBN 978-0-521-36865-0. • Katz, Robert (1964), Axiomatic Analysis, Boston, MA: D. C. Heath and Company. • Mendelson, Elliott (1997), Introduction to Mathematical Logic (4th ed.), London: Chapman & Hall, ISBN 978-0-412-80830-2. Mathematical logic 34 • Schwichtenberg, Helmut (2003—2004), Mathematical Logic (http://www.mathematik.uni-muenchen.de/ ~schwicht/lectures/logic/ws03/ml.pdf), Munich, Germany: Mathematisches Institut der Universitat Miinchen. • Shawn Hedman, A first course in logic: an introduction to model theory, proof theory, computability, and complexity, Oxford University Press, 2004, ISBN 0198529813. Covers logics in close reation with computability theory and complexity theory Graduate texts • Andrews, Peter B. (2002), An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof (2nd ed.), Boston: Kluwer Academic Publishers, ISBN 978-1-4020-0763-7. • Barwise, Jon, ed. (1982), Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, North Holland, ISBN 978-0-444-86388-1. • Hodges, Wilfrid (1997), A shorter model theory, Cambridge: Cambridge University Press, ISBN 978-0-521-58713-6. • (2003), Set Theory: Millennium Edition, Springer Monographs in Mathematics, Berlin, New York: Springer- Verlag, ISBN 978-3-540-44085-7. • Shoenfield, Joseph R. (2001) [1967], Mathematical Logic (2nd ed.), A K Peters, ISBN 978-1-56881-135-2. • ; Schwichtenberg, Helmut (2000), Basic Proof Theory, Cambridge Tracts in Theoretical Computer Science (2nd ed.), Cambridge: Cambridge University Press, ISBN 978-0-521-77911-1. Research papers, monographs, texts, and surveys • Cohen, P. J. (1966), Set Theory and the Continuum Hypothesis, Menlo Park, CA: W. A. Benjamin. • Davis, Martin (1973), "Hilbert's tenth problem is unsolvable" (http://jstor.org/stable/2318447), The American Mathematical Monthly (The American Mathematical Monthly, Vol. 80, No. 3) 80 (3): 233-269, doi: 10.2307/23 18447, reprinted as an appendix in Martin Davis, Computability and Unsolvability, Dover reprint 1982. JStor(http://links.jstor.org/sici?sici=0002-9890(197303)80:3<233:HTPIU>2.0.CO;2-E) • Felscher, Walter (2000), "Bolzano, Cauchy, Epsilon, Delta" (http://jstor.org/stable/2695743), The American Mathematical Monthly (The American Mathematical Monthly, Vol. 107, No. 9) 107 (9): 844-862, doi: 10.2307/2695743. JSTOR (http://links.jstor.org/sici?sici=0002-9890(200011)107:9<844:BCED>2.0. CO;2-L) • Ferreiros, Jose (2001), "The Road to Modern Logic-An Interpretation" (http://jstor.org/stable/2687794), Bulletin of Symbolic Logic (The Bulletin of Symbolic Logic, Vol. 7, No. 4) 7 (4): 441-484, doi: 10.2307/2687794. JStor(http://links.jstor.org/sici?sici=1079-8986(200112)7:4<441:TRTMLI>2.0.CO;2-O) • Hamkins, Joel David; Lowe, Benedikt, "The modal logic of forcing", Transactions of the American Mathematical Society, to appear. Electronic posting by the journal (http://www.ams.org/tran/0000-000-00/ S0002-9947-07-04297-3/home.html) • Katz, Victor J. (1998), A History of Mathematics, Addison-Wesley, ISBN 321 01618 1. • Morley, Michael (1965), "Categoricity in Power" (http://jstor.org/stable/1994188), Transactions of the American Mathematical Society (Transactions of the American Mathematical Society, Vol. 114, No. 2) 114 (2): 514-538, doi: 10.2307/1994188. • Soare, Robert I. (1996), "Computability and recursion" (http://jstor.org/stable/420992), Bulletin of Symbolic Logic (The Bulletin of Symbolic Logic, Vol. 2, No. 3) 2 (3): 284-321, doi: 10.2307/420992. • Solovay, Robert M. (1976), "Provability Interpretations of Modal Logic", Israel Journal of Mathematics 25: 287-304, doi:10.1007/BF02757006. • Woodin, W. Hugh (2001), "The Continuum Hypothesis, Part I", Notices of the American Mathematical Society 48 (6). PDF (http://www.ams.org/notices/200106/fea-woodin.pdf) Mathematical logic 35 Classical papers, texts, and collections • Burali-Forti, Cesare (1897), A question on transfinite numbers, reprinted in van Heijenoort 1976, pp. 104—1 11. • Dedekind, Richard (1872), Stetigkeit und irrationale Zahlen. English translation of title: "Consistency and irrational numbers". • Dedekind, Richard (1888), Was sind und was sollen die Zahlen? Two English translations: • 1963 (1901). Essays on the Theory of Numbers . Beman, W. W., ed. and trans. Dover. • 1996. In From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols, Ewald, William B., ed., Oxford University Press: 787—832. • Fraenkel, Abraham A. (1922), "Der Begriff 'definit' und die Unabhangigkeit des Auswahlsaxioms", Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, pp. 253—257 (German), reprinted in English translation as "The notion of 'definite' and the independence of the axiom of choice", van Heijenoort 1976, pp. 284—289. • Frege Gottlob (1879), Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a. S.: Louis Nebert. Translation: Concept Script, a formal language of pure thought modelled upon that of arithmetic, by S. Bauer-Mengelberg in Jean Van Heijenoort, ed., 1967. From Frege to Godel: A Source Book in Mathematical Logic, 1879—1931. Harvard University Press. • Frege Gottlob (1884), Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung iiber den Begriff der Zahl. Breslau: W. Koebner. Translation: J. L. Austin, 1974. The Foundations of Arithmetic: A logico -mathematical enquiry into the concept of number, 2nd ed. Blackwell. • Gentzen, Gerhard (1936), "Die Widerspruchsfreiheit der reinen Zahlentheorie", Mathematische Annalen \\2: 132—213, doi:10.1007/BF01565428, reprinted in English translation in Gentzen's Collected works, M. E. Szabo, ed., North-Holland, Amsterdam, 1969. • Godel, Kurt (1929), "Uber die Vollstdndigkeit des Logikkalkiils", doctoral dissertation, University Of Vienna. English translation of title: "Completeness of the logical calculus". • Godel, Kurt (1930), "Die Vollstandigkeit der Axiome des logischen Funktionen-kalkiils", Monatshefte fur Mathematik und Physik 37: 349-360, doi:10.1007/BF01696781. English translation of title: "The completeness of the axioms of the calculus of logical functions". • Godel, Kurt (1931), "Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme I", Monatshefte fur Mathematik und Physik 38 (1): 173-198, doi:10.1007/BF01700692, see On Formally Undecidable Propositions of Principia Mathematica and Related Systems for details on English translations. • Godel, Kurt (1958), "Uber eine bisher noch nicht beniitzte Erweiterung des finiten Standpunktes", Dialectica. International Journal of Philosophy 12: 280-287, doi:10.1 1 1 1/j. 1746-8361. 1958.tb01464.x, reprinted in English translation in Godel's Collected Works, vol II, Soloman Feferman et al., eds. Oxford University Press, 1990. • van Heijenoort, Jean, ed. (1967, 1976 3rd printing with corrections), From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931 (3rd ed.), Cambridge, Mass: Harvard University Press, ISBN 0-674-32449-8 (pbk.) • Hilbert, David (1899), Grundlagen der Geometrie, Leipzig: Teubner, English 1902 edition (The Foundations of Geometry) republished 1980, Open Court, Chicago. • David, Hilbert (1929), "Probleme der Grundlegung der Mathematik", Mathematische Annalen 102: 1—9, doi:10.1007/BF01782335. Lecture given at the International Congress of Mathematicians, 3 September 1928. Published in English translation as "The Grounding of Elementary Number Theory", in Mancosu 1998, pp. 266-273. • (1943), "Recursive Predicates and Quantifiers" (http://jstor.org/stable/1990131), American Mathematical Society Transactions (Transactions of the American Mathematical Society, Vol. 53, No. 1) 54 (1): 41—73, doi:10.2307/1990131. Mathematical logic 36 • Lobachevsky, Nikolai (1840), Geometrishe Untersuchungen zur Theorie der Parellellinien (German). Reprinted in English translation as "Geometric Investigations on the Theory of Parallel Lines" in Non-Euclidean Geometry, Robert Bonola (ed.), Dover, 1955. ISBN 0486600270 • (1915), "Uber Moglichkeiten im Relativkalkul", Mathematische Annalen 76: 447-470, doi:10.1007/BF01458217, ISSN 0025-5831 (German). Translated as "On possibilities in the calculus of relatives" in Jean van Heijenoort, 1967. A Source Book in Mathematical Logic, 1879—1931. Harvard Univ. Press: 228—251. • Mancosu, Paolo, ed. (1998), From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s, Oxford: Oxford University Press. • Pasch, Moritz (1882), Vorlesungen iiber neuere Geometric • Peano, Giuseppe (1888), Arithmetices principia, nova methodo exposita (Italian), excerpt reprinted in English stranslation as "The principles of arithmetic, presented by a new method", van Heijenoort 1976, pp. 83 97. • Richard, Jules (1905), "Les principes des mathematiques et le probleme des ensembles", Revue generale des sciences pures et appliquees 16: 541 (French), reprinted in English translation as "The principles of mathematics and the problems of sets", van Heijenoort 1976, pp. 142—144. • Skolem, Thoralf (1920), "Logisch-kombinatorische Untersuchungen iiber die Erfiillbarkeit oder Beweisbarkeit mathematischer Satze nebst einem Theoreme iiber dichte Mengen", Videnskapsselskapet Skrifter, I. Matematisk-naturvidenskabelig Klasse 6: 1—36. • Tarski, Alfred (1948), A decision method for elementary algebra and geometry, Santa Monica, California: RAND Corporation • Turing, Alan M. (1939), "Systems of Logic Based on Ordinals", Proceedings of the London Mathematical Society 45(2): 161-228, doi:10.1112/plms/s2-45.1. 161 • Zermelo, Ernst (1904), "Beweis, daB jede Menge wohlgeordnet werden kann", Mathematische Annalen 59: 514—516, doi:10.1007/BF01445300 (German), reprinted in English translation as "Proof that every set can be well-ordered", van Heijenoort 1976, pp. 139—141. • Zermelo, Ernst (1908a), "Neuer Beweis fur die Moglichkeit einer Wohlordnung", Mathematische Annalen 65: 107-128, doi:10.1007/BF01450054, ISSN 0025-5831 (German), reprinted in English translation as "A new proof of the possibility of a well-ordering", van Heijenoort 1976, pp. 183—198. • Zermelo, Ernst (1908b), "Untersuchungen iiber die Grundlagen der Mengenlehre", Mathematische Annalen 65: 261-281, doi:10.1007/BF01449999. External links • Mathematical Logic around the world (http://world.logic.at/) • Polyvalued logic (http://home.swipnet.se/~w-33552/logic/home/index.htm) • forall x: an introduction to formal logic (http://www.fecundity.com/logic/), by P.D. Magnus, is a free textbook. • A Problem Course in Mathematical Logic (http://euclid.trentu.ca/math/sb/pcml/), by Stefan Bilaniuk, is another free textbook. • Detlovs, Vilnis, and Podnieks, Karlis (University of Latvia) Introduction to Mathematical Logic. (http://www. ltn.lv/~podnieks/mlog/ml.htm) A hyper-textbook. • Stanford Encyclopedia of Philosophy: Classical Logic (http://plato.stanford.edu/entries/logic-classical/) - by Stewart Shapiro. • Stanford Encyclopedia of Philosophy: First-order Model Theory (http://plato.stanford.edu/entries/ modeltheory-fo/) — by Wilfrid Hodges. • The London Philosophy Study Guide (http://www.ucl.ac.uk/philosophy/LPSG/) offers many suggestions on what to read, depending on the student's familiarity with the subject: • Mathematical Logic (http://www.ucl.ac.uk/philosophy/LPSG/MathLogic.htm) • Set Theory & Further Logic (http://www.ucl.ac.uk/philosophy/LPSG/SetTheory.htm) Mathematical logic 37 • Philosophy of Mathematics (http://www.ucl.ac.uk/philosophy/LPSG/PhilMath.htm) Symbolic logic Symbolic logic may refer to: • First-order logic, a system of formal logic • Mathematical logic, a field of mathematics Metalogic Metalogic is the study of the metatheory of logic. While logic is the study of the manner in which logical systems can be used to decide the correctness of arguments, metalogic studies the properties of the logical systems themselves. According to Geoffrey Hunter, while logic concerns itself with the "truths of logic," metalogic concerns itself with the theory of "sentences used to express truths of logic." The basic objects of study in metalogic are formal languages, formal systems, and their interpretations. The study of interpretation of formal systems is the branch of mathematical logic known as model theory, while the study of deductive apparatus is the branch known as proof theory. History Metalogical questions have been asked since the time of Aristotle. However, it was only with the rise of formal languages in the late 19th and early 20th century that investigations into the foundations of logic began to flourish. In 1904, David Hilbert observed that in investigating the foundations of mathematics that logical notions are presupposed, and therefore a simultaneous account of metalogical and metamathematical principles was required. Today, metalogic and metamathematics are largely synonymous with each other, and both have been substantially subsumed by mathematical logic in academia. Important distinctions in metalogic Metalanguage-Object language In metalogic, formal languages are sometimes called object languages. The language used to make statements about an object language is called a metalanguage. This distinction is a key difference between logic and metalogic. While logic deals with proofs in a formal system, expressed in some formal language, metalogic deals with proofs about a formal system which are expressed in a metalanguage about some object language. Syntax-semantics In metalogic, 'syntax' has to do with formal languages or formal systems without regard to any interpretation of them, whereas, 'semantics' has to do with interpretations of formal languages. The term 'syntactic' has a slightly wider scope than 'proof- theoretic', since it may be applied to properties of formal languages without any deductive systems, as well as to formal systems. 'Semantic' is synonymous with 'model-theoretic'. Metalogic 38 Use— mention In metalogic, the words 'use' and 'mention', in both their noun and verb forms, take on a technical sense in order to identify an important distinction. The use— mention distinction (sometimes referred to as the words-as-words distinction) is the distinction between using a word (or phrase) and mentioning it. Usually it is indicated that an expression is being mentioned rather than used by enclosing it in quotation marks, printing it in italics, or setting the expression by itself on a line. The enclosing in quotes of an expression gives us the name of an expression, for example: 'Metalogic' is the name of this article. This article is about metalogic. Type-token The type-token distinction is a distinction in metalogic, that separates an abstract concept from the objects which are particular instances of the concept. For example, the particular bicycle in your garage is a token of the type of thing known as "The bicycle." Whereas, the bicycle in your garage is in a particular place at a particular time, that is not true of "the bicycle" as used in the sentence: "The bicycle has become more popular recently." This distinction is used to clarify the meaning of symbols of formal languages. Overview Formal language A formal language is an organized set of symbols the essential feature of which is that it can be precisely defined in terms of just the shapes and locations of those symbols. Such a language can be defined, then, without any reference to any meanings of any of its expressions; it can exist before any interpretation is assigned to it — that is, before it has any meaning. First order logic is expressed in some formal language. A formal grammar determines which symbols and sets of symbols are formulas in a formal language. A formal language can be defined formally as a set A of strings (finite sequences) on a fixed alphabet a. Some authors, including Carnap, define the language as the ordered pair <a, A>. Carnap also requires that each element of a must occur in at least one string in A. Formation rules Formation rules (also called formal grammar) are a precise description of the well-formed formulas of a formal language. It is synonymous with the set of strings over the alphabet of the formal language which constitute well formed formulas. However, it does not describe their semantics (i.e. what they mean). Formal systems A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules (also called inference rules) or a set of axioms, or have both. A formal system is used to derive one expression from one or more other expressions. A formal system can be formally defined as an ordered triple <a, 2? . T) d>, where J) d is the relation of direct derivability. This relation is understood in a comprehensive sense such that the primitive sentences of the formal system are taken as directly derivable from the empty set of sentences. Direct derivability is a relation between a sentence and a finite, possibly empty set of sentences. Axioms are laid down in such a way that every first place member of J) d is a member of 2T an d every second place member is a finite subset of J • Metalogic 39 It is also possible to define a formal system using only the relation J) d. In this way we can omit 2T > and a in the definitions of interpreted formal language, and interpreted formal system. However, this method can be more T31 difficult to understand and work with. Formal proofs A formal proof is a sequence of well-formed formulas of a formal language, the last one of which is a theorem of a formal system. The theorem is a syntactic consequence of all the well formed formulae preceding it in the proof. For a well formed formula to qualify as part of a proof, it must be the result of applying a rule of the deductive apparatus of some formal system to the previous well formed formulae in the proof sequence. Interpretations An interpretation of a formal system is the assignment of meanings, to the symbols, and truth-values to the sentences of the formal system. The study of interpretations is called Formal semantics. Giving an interpretation is synonymous with constructing a model. Results in metalogic Results in metalogic consist of such things as formal proofs demonstrating the consistency, completeness, and decidability of particular formal systems. Major results in metalogic include: Proof of the uncountability of the set of all subsets of the set of natural numbers (Cantor's theorem 1891) Lowenheim-Skolem theorem (Leopold Lowenheim 1915 and Thoralf Skolem 1919) Proof of the consistency of truth-functional propositional logic (Emil Post 1920) mi Proof of the semantic completeness of truth-functional propositional logic (Paul Bernays 1918), (Emil Post 1920) [2] T21 Proof of the syntactic completeness of truth-functional propositional logic (Emil Post 1920) T21 Proof of the decidability of truth-functional propositional logic (Emil Post 1920) Proof of the consistency of first order monadic predicate logic (Leopold Lowenheim 1915) Proof of the semantic completeness of first order monadic predicate logic (Leopold Lowenheim 1915) Proof of the decidability of first order monadic predicate logic (Leopold Lowenheim 1915) Proof of the consistency of first order predicate logic (David Hilbert and Wilhelm Ackermann 1928) Proof of the semantic completeness of first order predicate logic (Godel's completeness theorem 1930) Proof of the undecidability of first order predicate logic (Church's theorem 1936) Godel's first incompleteness theorem 1931 Godel's second incompleteness theorem 1931 See also • Metamathematics References [1] Harry J. Gensler, Introduction to Logic, Routledge, 2001, p. 253. [2] Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Press, 197 1 [3] Rudolf Carnap (1958) Introduction to Symbolic Logic and its Applications, p. 102. [4] Hao Wang, Reflections on Kurt Godel Metatheory 40 Metatheory A metatheory or meta-theory is a theory whose subject matter is some other theory. In other words it is a theory about a theory. Statements made in the metatheory about the theory are called metatheorems. According to the systemic TOGA meta-theory , a meta-theory may refer to the specific point of view on a theory and to its subjective meta-properties, but not to its application domain. In the above sense, a theory T of the domain D is a meta-theory if D is a theory or a set of theories. A general theory is not a meta-theory because its domain D are not theories. The following is an example of a meta-theoretical statement: Any physical theory is always provisional, in the sense that it is only a hypothesis; you can never prove it. No matter how many times the results of experiments agree with some theory, you can never be sure that the next time the result will not contradict the theory. On the other hand, you can disprove a theory by finding even a single observation that disagrees with the predictions of the theory. Meta-theory belongs to the philosophical specialty of epistemology and metamathematics, as well as being an object of concern to the area in which the individual theory is conceived. An emerging domain of meta-theories is systemics. Taxonomy Examining groups of related theories, a first finding may be to identify classes of theories, thus specifying a taxonomy of theories. A proof engendered by a metatheory is called a metatheorem. History The concept burst upon the scene of twentieth-century philosophy as a result of the work of the German mathematician David Hilbert, who in 1905 published a proposal for proof of the consistency of mathematics, creating the field of metamathematics. His hopes for the success of this proof were dashed by the work of Kurt Godel who in 1931 proved this to be unattainable by his incompleteness theorems. Nevertheless, his program of unsolved mathematical problems, out of which grew this metamathematical proposal, continued to influence the direction of mathematics for the rest of the twentieth century. The study of metatheory became widespread during the rest of that century by its application in other fields, notably scientific linguistics and its concept of metalanguage. See also • meta- • meta-knowledge • Metalogic • Metamathematics • Metahistory, a book by Hayden White • Philosophy of social science Metatheory 4 1 References [1] * Meta-Knowledge Unified Framework (http://hid.casaccia.enea.it/Meta-know-l.htm) - the TOGA meta-theory [2] Stephen Hawking in A Brief History of Time External links • Meta-theoretical Issues (2003), Lyle Flint (http://www.bsu.edu/classes/flint/comm360/metatheo.html) Metamathematics Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Metamathematical metatheorems about mathematics itself were originally differentiated from ordinary mathematical theorems in the 19th century, to focus on what was then called the foundational crisis of mathematics. Richard's paradox (Richard 1905) concerning certain 'definitions' of real numbers in the English language is an example of the sort of contradictions which can easily occur if one fails to distinguish between mathematics and metamathematics. The term "metamathematics" is sometimes used as a synonym for certain elementary parts of formal logic, including propositional logic and predicate logic. History Metamathematics was intimately connected to mathematical logic, so that the early histories of the two fields, during the late 19th and early 20th centuries, largely overlap. More recently, mathematical logic has often included the study of new pure mathematics, such as set theory, recursion theory and pure model theory, which is not directly related to metamathematics. Serious metamathematical reflection began with the work of Gottlob Frege, especially his Begriffsschrift. David Hilbert was the first to invoke the term "metamathematics" with regularity (see Hilbert's program). In his hands, it meant something akin to contemporary proof theory, in which finitary methods are used to study various axiomatized mathematical theorems. Other prominent figures in the field include Bertrand Russell, Thoralf Skolem, Emil Post, Alonzo Church, Stephen Kleene, Willard Quine, Paul Benacerraf, Hilary Putnam, Gregory Chaitin, Alfred Tarski and Kurt Godel. In particular, arguably the greatest achievement of metamathematics and the philosophy of mathematics to date is Godel's incompleteness theorem: proof that given any finite number of axioms for Peano arithmetic, there will be true statements about that arithmetic that cannot be proved from those axioms. Milestones • Principia Mathematica (Whitehead and Russell 1925) • Godel's completeness theorem, 1930 • Godel's incompleteness theorem, 1931 • Tarski's definition of model-theoretic satisfaction, now called the T-schema • The proof of the impossibility of the Entscheidungsproblem, obtained independently in 1936—1937 by Church and Turing. Metamathematics 42 See also • Meta- • Model theory • Philosophy of mathematics • Proof theory References • W. J. Blok and Don Pigozzi, "Alfred Tarski's Work on General Metamathematics", The Journal of Symbolic Logic, v. 53, No. 1 (Mar., 1988), pp. 36-50. • I. J. Good. "A Note on Richard's Paradox". Mind, New Series, Vol. 75, No. 299 (Jul., 1966), p. 431. JStor [1] • Douglas Hofstadter, 1980. Godel, Escher, Bach. Vintage Books. Aimed at laypeople. • Stephen Cole Kleene, 1952. Introduction to Metamathematics. North Holland. Aimed at mathematicians. • Jules Richard, Les Principes des Mathematiques et le Probleme des Ensembles, Revue Generale des Sciences Pures et Appliquees (1905); translated in Heijenoort J. van (ed.), Source Book in Mathematical Logic 1879-1931 (Cambridge, Mass., 1964). • Alfred North Whitehead, and Bertrand Russell. Principia Mathematica, 3 vols, Cambridge University Press, 1910, 1912, and 1913. Second edition, 1925 (Vol. 1), 1927 (Vols 2, 3). Abridged as Principia Mathematica to *56, Cambridge University Press, 1962. References [1] http://www.jstor.org/stable/2251836 43 Abstract Algebra Abstract algebra Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and real or complex numbers, often now called elementary algebra. The distinction is rarely made in more recent writings. Contemporary mathematics and mathematical physics make extensive use of abstract algebra; for example, theoretical physics draws on Lie algebras. Subject areas such as algebraic number theory, algebraic topology, and algebraic geometry apply algebraic methods to other areas of mathematics. Representation theory, roughly speaking, takes the 'abstract' out of 'abstract algebra', studying the concrete side of a given structure; see model theory. Two mathematical subject areas that study the properties of algebraic structures viewed as a whole are universal algebra and category theory. Algebraic structures, together with the associated homomorphisms, form categories. Category theory is a powerful formalism for studying and comparing different algebraic structures. History and examples As in other parts of mathematics, concrete problems and examples have played important roles in the development of algebra. Through the end of the nineteenth century many, perhaps most of these problems were in some way related to the theory of algebraic equations. Among major themes we can mention: • solving of systems of linear equations, which led to matrices, determinants and linear algebra. • attempts to find formulae for solutions of general polynomial equations of higher degree that resulted in discovery of groups as abstract manifestations of symmetry; • and arithmetical investigations of quadratic and higher degree forms and diophantine equations, notably, in proving Fermat's last theorem, that directly produced the notions of a ring and ideal. Numerous textbooks in abstract algebra start with axiomatic definitions of various algebraic structures and then proceed to establish their properties, creating a false impression that somehow in algebra axioms had come first and then served as a motivation and as a basis of further study. The true order of historical development was almost exactly the opposite. For example, the hypercomplex numbers of the nineteenth century had kinematic and physical motivations but challenged comprehension. Most theories that we now recognize as parts of algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts. An archetypical example of this progressive synthesis can be seen in the theory of groups. Abstract algebra 44 Early group theory There were several threads in the early development of group theory, in modern language loosely corresponding to number theory, theory of equations, and geometry, of which we concentrate on the first two. Leonhard Euler considered algebraic operations on numbers modulo an integer, modular arithmetic, proving his generalization of Fermat's little theorem. These investigations were taken much further by Carl Friedrich Gauss, who considered the structure of multiplicative groups of residues mod n and established many properties of cyclic and more general abelian groups that arise in this way. In his investigations of composition of binary quadratic forms, Gauss explicitly stated the associative law for the composition of forms, but like Euler before him, he seems to have been more interested in concrete results than in general theory. In 1870, Leopold Kronecker gave a definition of an abelian group in the context of ideal class groups of a number field, a far-reaching generalization of Gauss's work. It appears that he did not tie it with previous work on groups, in particular, permutation groups. In 1882 considering the same question, Heinrich M. Weber realized the connection and gave a similar definition that involved the cancellation property but omitted the existence of the inverse element, which was sufficient in his context (finite groups). Permutations were studied by Joseph Lagrange in his 1770 paper Reflexions sur la resolution algebrique des equations devoted to solutions of algebraic equations, in which he introduced Lagrange resolvents. Lagrange's goal was to understand why equations of third and fourth degree admit formulae for solutions, and he identified as key objects permutations of the roots. An important novel step taken by Lagrange in this paper was the abstract view of the roots, i.e. as symbols and not as numbers. However, he did not consider composition of permutations. Serendipitously, the first edition of Edward Waring's Meditationes Algebraicae appeared in the same year, with an expanded version published in 1782. Waring proved the main theorem on symmetric functions, and specially considered the relation between the roots of a quartic equation and its resolvent cubic. Memoire sur la resolution des equations of Alexandre Vandermonde (1771) developed the theory of symmetric functions from a slightly different angle, but like Lagrange, with the goal of understanding solvability of algebraic equations. Kronecker claimed in 1888 that the study of modern algebra began with this first paper of Vandermonde. Cauchy states quite clearly that Vandermonde had priority over Lagrange for this remarkable idea which eventually led to the study of group theory. Paolo Ruffini was the first person to develop the theory of permutation groups, and like his predecessors, also in the context of solving algebraic equations. His goal was to establish impossibility of algebraic solution to a general algebraic equation of degree greater than four. En route to this goal he introduced the notion of the order of an element of a group, conjugacy, the cycle decomposition of elements of permutation groups and the notions of primitive and imprimitive and proved some important theorems relating these concepts, such as if G is a subgroup of S whose order is divisible by 5 then G contains an element of order 5. Note, however, that he got by without formalizing the concept of a group, or even of a permutation group. The next step was taken by Evariste Galois in 1832, although his work remained unpublished until 1846, when he considered for the first time what we now call the closure property of a group of permutations, which he expressed as ... if in such a group one has the substitutions S and T then one has the substitution ST. The theory of permutation groups received further far-reaching development in the hands of Augustin Cauchy and Camille Jordan, both through introduction of new concepts and, primarily, a great wealth of results about special classes of permutation groups and even some general theorems. Among other things, Jordan defined a notion of isomorphism, still in the context of permutation groups and, incidentally, it was he who put the term group in wide use. The abstract notion of a group appeared for the first time in Arthur Cay ley's papers in 1854. Cayley realized that a group need not be a permutation group (or even finite), and may instead consist of matrices, whose algebraic properties, such as multiplication and inverses, he systematically investigated in succeeding years. Much later Abstract algebra 45 Cayley would revisit the question whether abstract groups were more general than permutation groups, and establish that, in fact, any group is isomorphic to a group of permutations. Modern algebra The end of 19th and the beginning of the 20th century saw a tremendous shift in methodology of mathematics. No longer satisfied with establishing properties of concrete objects, mathematicians started to turn their attention to general theory. For example, results about various groups of permutations came to be seen as instances of general theorems that concern a general notion of an abstract group. Questions of structure and classification of various mathematical objects came to forefront. These processes were occurring throughout all of mathematics, but became especially pronounced in algebra. Formal definition through primitive operations and axioms were proposed for many basic algebraic structures, such as groups, rings, and fields. The algebraic investigations of general fields by Ernst Steinitz and of commutative and then general rings by David Hilbert, Emil Artin and Emmy Noether, building up on the work of Ernst Kummer, Leopold Kronecker and Richard Dedekind, who had considered ideals in commutative rings, and of Georg Frobenius and Issai Schur, concerning representation theory of groups, came to define abstract algebra. These developments of the last quarter of the 19th century and the first quarter of 20th century were systematically exposed in B artel van der Waerden's Moderne algebra, the two-volume monograph published in 1930—1931 that forever changed for the mathematical world the meaning of the word algebra from the theory of equations to the theory of algebraic structures. An example Abstract algebra facilitates the study of properties and patterns that seemingly disparate mathematical concepts have in common. For example, consider the distinct operations of function composition, f(g(x)), and of matrix multiplication, AB. These two operations have, in fact, the same structure. To see this, think about multiplying two square matrices, AB, by a one column vector, x. This defines a function equivalent to composing Ay with Bx: Ay = A(Bx) = (AB)x. Functions under composition and matrices under multiplication are examples of monoids. A set S and a binary operation over S, denoted by concatenation, form a monoid if the operation associates, (ab)c = a(bc), and if there exists an e€S, such that ae = ea = a. Notes [1] Vandermonde biography in Mac Tutor History of Mathematics Archive (http://www-groups.dcs.st-and.ac.uk/~history/Biographies/ Vandermonde. html) . References • Allenby, R.B.J.T. (1991), Rings, Fields and Groups, Butterworth-Heinemann, ISBN 978-0-340-54440-2 • Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1 • Burris, Stanley N; Sankappanavar, H. P. (1999) [1981], A Course in Universal Algebra (http://www.math. uwaterloo.ca/~snburris/htdocs/ualg.html) • Gilbert, Jimmie; Gilbert, Linda (2005), Elements of Modern Algebra, Thomson Brooks/Cole, ISBN 978-0-534-40264-8 • Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer- Verlag, MR1878556, ISBN 978-0-387-95385-4 • Sethuraman, B. A. (1996), Rings, Fields, Vector Spaces, and Group Theory: An Introduction to Abstract Algebra via Geometric Constructibility, Berlin, New York: Springer- Verlag, ISBN 978-0-387-94848-5 • Whitehead, C. (2002), Guide to Abstract Algebra (2nd ed.), Houndmills: Palgrave, ISBN 978-0-333-79447-0 • W. Keith Nicholson, Introduction to abstract algebra • John R. Durbin, Modern algebra : an introduction Abstract algebra 46 • Raymond A. Barnett, Intermediate algebra; structure and use External links • John Beachy: Abstract Algebra On Line (http://www.math.niu.edu/~beachy/aaol/contents.html), Comprehensive list of definitions and theorems. • Edwin Connell " Elements of Abstract and Linear Algebra (http://www.math.miami.edu/~ec/book/)", Free online textbook. • Fredrick M. Goodman: Algebra: Abstract and Concrete (http://www.math.uiowa.edu/~goodman/ algebrabook. dir/algebrabook. html) . • Judson, Thomas W. (1997), Abstract Algebra: Theory and Applications (http://abstract.ups.edu) An introductory undergraduate text in the spirit of texts by Gallian or Herstein, covering groups, rings, integral domains, fields and Galois theory. Free downloadable PDF with open-source GFDL license. Universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes "the theory of groups" as an object of study. Basic idea From the point of view of universal algebra, an algebra (or algebraic structure) is a set A together with a collection of operations on A. An n-ary operation on A is a function that takes n elements of A and returns a single element of A. Thus, a O-ary operation (or nullary operation) can be represented simply as an element of A, or a constant, often denoted by a letter like a. A 1-ary operation (or unary operation) is simply a function from A to A, often denoted by a symbol placed in front of its argument, like ~x. A 2-ary operation (or binary operation) is often denoted by a symbol placed between its arguments, like x * y. Operations of higher or unspecified arity are usually denoted by function symbols, with the arguments placed in parentheses and separated by commas, like f(x,y,z) or f(x ,...,x ). Some researchers allow infinitary operations, such as f\ ae j X a where / is an infinite index set, thus leading into the algebraic theory of complete lattices. One way of talking about an algebra, then, is by referring to it as an algebra of a certain type Q, where Qis an ordered sequence of natural numbers representing the arity of the operations of the algebra. Equations After the operations have been specified, the nature of the algebra can be further limited by axioms, which in universal algebra often take the form of identities, or equational laws. An example is the associative axiom for a binary operation, which is given by the equation x * (y * z) = (x * y) * z. The axiom is intended to hold for all elements x, y, and z of the set A. Varieties An algebraic structure which can be defined by identities is called a variety, and these are sufficiently important that some authors consider varieties the only object of study in universal algebra, while others consider them an object. Restricting one's study to varieties rules out: • Predicate logic, notably quantification, including existential quantification ( 3 ) an d universal quantification ( \f ) Universal algebra 47 • Relations, including inequalities, both a ^ b and order relations In this narrower definition, universal algebra can be seen as a special branch of model theory, in which we are typically dealing with structures having operations only (i.e. the type can have symbols for functions but not for relations other than equality), and in which the language used to talk about these structures uses equations only. Not all algebraic structures in a wider sense fall into this scope. For example ordered groups are not studied in mainstream universal algebra because they involve an ordering relation. A more fundamental restriction is that universal algebra cannot study the class of fields, because there is no type in which all field laws can be written as equations (inverses of elements are defined for all non-zero elements in a field, so inversion cannot simply be added to the type). One advantage of this restriction is that the structures studied in universal algebra can be defined in any category which has finite products. For example, a topological group is just a group in the category of topological spaces. Examples Most of the usual algebraic systems of mathematics are examples of varieties, but not always in an obvious way — the usual definitions often involve quantification or inequalities. Groups To see how this works, let's consider the definition of a group. Normally a group is defined in terms of a single binary operation *, subject to these axioms: • Associativity (as in the previous section): x * (y * z) = (x * y) * z. • Identity element: There exists an element e such that for each element x, e * x = x = x* e. • Inverse element: It can easily be seen that the identity element is unique. If we denote this unique identity element by e then for each x, there exists an element i such that x * i - e - i * x. (Sometimes you will also see an axiom called "closure", stating that x * y belongs to the set A whenever x and y do. But from a universal algebraist's point of view, that is already implied when you call * a binary operation.) Now, this definition of a group is problematic from the point of view of universal algebra. The reason is that the axioms of the identity element and inversion are not stated purely in terms of equational laws but also have clauses involving the phrase "there exists ... such that ...". This is inconvenient; the list of group properties can be simplified to universally quantified equations if we add a nullary operation e and a unary operation ~ in addition to the binary operation *, then list the axioms for these three operations as follows: • Associativity: x * (y * z) = (x * y) * z. • Identity element: e*x = x = x*e. • Inverse element: x * (~x) = e = (~x) * x. (Of course, we usually write "x ~ " instead of "~x", which shows that the notation for operations of low arity is not always as given in the second paragraph.) What has changed is that in the usual definition there are: • a single binary operation (signature (2)) • 1 equational law (associativity) • 2 quantified laws (identity and inverse) ...while in the universal algebra definition there are • 3 operations: one binary, one unary, and one nullary (signature (2,1,0)) • 3 equational laws (associativity, identity, and inverse) • no quantified laws Universal algebra 48 It's important to check that this really does capture the definition of a group. The reason that it might not is that specifying one of these universal groups might give more information than specifying one of the usual kind of group. After all, nothing in the usual definition said that the identity element e was unique; if there is another identity element e', then it's ambiguous which one should be the value of the nullary operator e. However, this is not a problem because identity elements can be proved to be always unique. The same thing is true of inverse elements. So the universal algebraist's definition of a group really is equivalent to the usual definition. Basic constructions We assume that the type, fl, has been fixed. Then there are three basic constructions in universal algebra: homomorphic image, subalgebra, and product. A homomorphism between two algebras A and B is a function h: A — > B from the set A to the set B such that, for every operation / (of arity, say, n), h(f (x ,...,x )) = f„(h(x ),..., h(x )). (Here, subscripts are placed on f to indicate A L fl D L Yl whether it is the version of /in A or B. In theory, you could tell this from the context, so these subscripts are usually left off.) For example, if e is a constant (nullary operation), then h(e ) = e . If - is a unary operation, then h{~x) = A B ~h{x). If * is a binary operation, then h(x * y) = h{x) * h(y). And so on. A few of the things that can be done with homomorphisms, as well as definitions of certain special kinds of homomorphisms, are listed under the entry Homomorphism. In particular, we can take the homomorphic image of an algebra, h{A). A subalgebra of A is a subset of A that is closed under all the operations of A. A product of some set of algebraic structures is the cartesian product of the sets with the operations defined coordinatewise. Some basic theorems • The Isomorphism theorems, which encompass the isomorphism theorems of groups, rings, modules, etc. • Birkhoff s HSP Theorem, which states that a class of algebras is a variety if and only if it is closed under homomorphic images, subalgebras, and arbitrary direct products. Motivations and applications In addition to its unifying approach, universal algebra also gives deep theorems and important examples and counterexamples. It provides a useful framework for those who intend to start the study of new classes of algebras. It can enable the use of methods invented for some particular classes of algebras to other classes of algebras, by recasting the methods in terms of universal algebra (if possible), and then interpreting these as applied to other classes. It has also provided conceptual clarification; as J.D.H. Smith puts it, "What looks messy and complicated in a particular framework may turn out to be simple and obvious in the proper general one. " In particular, universal algebra can be applied to the study of monoids, rings, and lattices. Before universal algebra came along, many theorems (most notably the isomorphism theorems) were proved separately in all of these fields, but with universal algebra, they can be proven once and for all for every kind of algebraic system. Category theory and operads A more generalised programme along these lines is carried out by category theory. Given a list of operations and axioms in universal algebra, the corresponding algebras and homomorphisms are the objects and morphisms of a category. Category theory applies to many situations where universal algebra does not, extending the reach of the theorems. Conversely, many theorems that hold in universal algebra do not generalise all the way to category theory. Thus both fields of study are useful. A more recent development in category theory that generalizes operations is operad theory — an operad is a set of operations, similar to a universal algebra. Universal algebra 49 History In Alfred North Whitehead's book A Treatise on Universal Algebra, published in 1898, the term universal algebra had essentially the same meaning that it has today. Whitehead credits William Rowan Hamilton and Augustus De Morgan as originators of the subject matter, and James Joseph Sylvester with coining the term itself . At the time structures such as Lie algebras and hyperbolic quaternions drew attention to the need to expand algebraic structures beyond the associatively multiplicative class. In a review Alexander Macfarlane wrote: "The main idea of the work is not unification of the several methods, nor generalization of ordinary algebra so as to include them, but rather the comparative study of their several structures." At the time George Boole's algebra of logic made a strong counterpoint to ordinary number algebra, so the term "universal" served to calm strained sensibilities. Whitehead's early work sought to unify quaternions (due to Hamilton), Grassmann's Ausdehnungslehre, and Boole's algebra of logic. Whitehead wrote in his book: "Such algebras have an intrinsic value for separate detailed study; also they are worthy of comparative study, for the sake of the light thereby thrown on the general theory of symbolic reasoning, and on algebraic symbolism in particular. The comparative study necessarily presupposes some previous separate study, T21 comparison being impossible without knowledge. Whitehead, however, had no results of a general nature. Work on the subject was minimal until the early 1930s, when Garrett Birkhoff and 0ystein Ore began publishing on universal algebras. Developments in metamathematics and category theory in the 1940s and 1950s furthered the field, particularly the work of Abraham Robinson, Alfred Tarski, Andrzej Mostowski, and their students (Brainerd 1967). In the period between 1935 and 1950, most papers were written along the lines suggested by Birkhoffs papers, dealing with free algebras, congruence and subalgebra lattices, and homomorphism theorems. Although the development of mathematical logic had made applications to algebra possible, they came about slowly; results published by Anatoly Maltsev in the 1940s went unnoticed because of the war. Tarski's lecture at the 1950 International Congress of Mathematicians in Cambridge ushered in a new period in which model-theoretic aspects were developed, mainly by Tarski himself, as well as C.C. Chang, Leon Henkin, Bjarni Jonsson, R. C. Lyndon, and others. T31 In the late 1950s, E. Marczewski emphasized the importance of free algebras, leading to the publication of more than 50 papers on the algebraic theory of free algebras by Marczewski himself, together with J. Mycielski, W. Narkiewicz, W. Nitka, J. Plonka, S. Swierczkowski, K. Urbanik, and others. Footnotes [1] Gratzer, George. Universal Algebra, Van Nostrand Co., Inc., 1968, p. v. [2] Quoted in Gratzer, George. Universal Algebra, Van Nostrand Co., Inc., 1968. [3] Marczewski, E. "A general scheme of the notions of independence in mathematics." Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 6 (1958), 731-736. References • Bergman, George M., 1998. An Invitation to General Algebra and Universal Constructions (http://math. berkeley.edu/~gbergman/245/) (pub. Henry Helson, 15 the Crescent, Berkeley CA, 94708) 398 pp. ISBN 0-9655211-4-1. • Birkhoff, Garrett, 1946. Universal algebra. Comptes Rendus du Premier Congres Canadien de Mathematiques, University of Toronto Press, Toronto, pp. 310—326. • Brainerd, Barron, Aug— Sep 1967. Review of Universal Algebra by P. M. Cohn. American Mathematical Monthly, 74(7): 878-880. Universal algebra 50 • Burris, Stanley N., and H.P. Sankappanavar, 1981. A Course in Universal Algebra (http://www.thoralf. uwaterloo.ca/htdocs/ualg.html) Springer- Verlag. ISBN 3-540-90578-2 Free online edition. • Cohn, Paul Moritz, 1981. Universal Algebra. Dordrecht , Netherlands: D.Reidel Publishing. ISBN 90-277-1213-1 (First published in 1965 by Harper & Row) • Freese, Ralph, and Ralph McKenzie, 1987. Commutator Theory for Congruence Modular Varieties (http://www. math.hawaii.edu/~ralph/Commutator), 1st ed. London Mathematical Society Lecture Note Series, 125. Cambridge Univ. Press. ISBN 0-521-34832-3. Free online second edition. • Gratzer, George, 1968. Universal Algebra D. Van Nostrand Company, Inc. • Hobby, David, and Ralph McKenzie, 1988. The Structure of Finite Algebras (http://www.ams.org/online_bks/ conm76) American Mathematical Society. ISBN 0-8218-3400-2. Free online edition. • Jipsen, Peter, and Henry Rose, 1992. Varieties of Lattices (http://wwwl.chapman.edu/~jipsen/ JipsenRoseVoL.html), Lecture Notes in Mathematics 1533. Springer Verlag. ISBN 0-387-56314-8. Free online edition. • Pigozzi, Don. General Theory of Algebras (http://bigcheese.math.sc.edu/~mcnulty/alglatvar/pigozzinotes. pdf). • Smith, J.D.H., 1976. Mal'cev Varieties, Springer- Verlag. • Whitehead, Alfred North, 1898. A Treatise on Universal Algebra (http://historical.library.cornell.edu/cgi-bin/ cul.math/docviewer?did=01950001&seq=5), Cambridge. {Mainly of historical interest.) External links • Algebra Universalis (http://www.birkhauser.ch/AU) — a journal dedicated to Universal Algebra. Heyting algebra In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras, named after Arend Heyting. Heyting algebras arise as models of intuitionistic logic, a logic in which the law of excluded middle does not in general hold. Complete Heyting algebras are a central object of study in pointless topology. Formal definition A Heyting algebra H is a bounded lattice such that for all a and binH there is a greatest element x of H such that aAi<ii. This element is the relative pseudo-complement of a with respect to b, and is denoted a — > b. We write 1 and for the largest and the smallest element of H, respectively. In any Heyting algebra, one defines the pseudo-complement -uc of any element x by setting ->x = (x — > 0). By definition, a a ->a - 0, and ->a is the largest element having this property. However, it is not in general true that a v -ifl = 1, thus -i is only a pseudo-complement, not a true complement, as would be the case in a Boolean algebra. A complete Heyting algebra is a Heyting algebra that is a complete lattice. A subalgebra of a Heyting algebra H is a subset H of H containing and 1 and closed under the operations a, v and — >. It follows that it is also closed under ->. A subalgebra is made into a Heyting algebra by the induced operations. Hey ting algebra 5 1 Alternative definitions Lattice-theoretic definitions An equivalent definition of Heyting algebras can be given by considering the mappings f a :H^H defined by f a (x) — a Ax, for some fixed a in H. A bounded lattice H is a Heyting algebra if and only if all mappings/ are the lower adjoint of a monotone Galois connection. In this case the respective upper adjoints g are given by g {x) = a — » x, where — > is defined as above. Yet another definition is as a residuated lattice whose monoid operation is a. The monoid unit must then be the top element 1 . Commutativity of this monoid implies that the two residuals coincide as a — > b. Bounded lattice with an implication operation Given a bounded lattice A with largest and smallest elements 1 and 0, and a binary operation — >, these together form a Heyting algebra if and only if the following hold: 1- a — » a = 1 2. aA(a—>b) = aAb 3. b A (a -> 6) = b 4. o^(6Ac) = (tt^t)A(a^c) where 4 is the distributive law for — >. Characterization using the axioms of intuitionistic logic This characterization of Heyting algebras makes the proof of the basic facts concerning the relationship between intuitionist propositional calculus and Heyting algebras immediate. (For these facts, see the sections "Provable identities" and "Universal constructions.") One should think of the element 1 as meaning, intuitively, "provably true." Compare with the axioms at Intuitionistic logic#Axiomatization. Given a set A with three binary operations — >, a and v, and two distinguished elements and 1, then A is a Heyting algebra for these operations (and the relation < defined by the condition that a < b when a — > b = 1) if and only if the following conditions hold for any elements x, y and z of A: = 1 then x = y, V) ~> ( x -»■ z )) = !> 1. lix -^y = - 1 and y — > x - 2. If 1 -^y = ■ 1, then y = 1 3. X —5 >(</- x) = l, 4. (x- -{y- > *)) - ((* - 5. x A y ^x = 1, 6. x A y^y = 1, 7. X —5 •(*/- Mj/)) = 1, 8. X —5 > xV y = 1, 9. y -> ■ x V y = 1, 10 • (* -*)- -+((v->*)- 4 11 • 0- -^ x = 1. (xVy^z)) = l, Finally, we define -a to be x — > 0. Condition 1 says that equivalent formulas should be identified. Condition 2 says that provably true formulas are closed under modus ponens. Conditions 3 and 4 are then conditions. Conditions 5, 6 and 7 are and conditions. Conditions 8, 9 and 10 are or conditions. Condition 1 1 is a. false condition. Of course, if a different set of axioms were chosen for logic, we could modify ours accordingly. Heyting algebra 52 (((^^p- p) - (pv -p)) - (-pv -.-. p)) v((^-.p-p)-(pv^p)) ((— p - p) - (pv -p)) -.(-.pv-.- J((-.-.p- p) - (pv-.pl) - (-.pv -.-.p)) - (-"PV (-'-'P - p» "fvl-.-. P-P) Examples • Every Boolean algebra is a Heyting algebra, with /? — > (7 given by -p v <?. • Every totally ordered set that is a bounded lattice is also a Heyting algebra, where p — > q is equal to q when /?><?, and 1 otherwise. • The simplest Heyting algebra that is not already a Boolean algebra is the totally ordered set {0, Vi, 1 } with — > defined as above. Notice that Vi v -tyi = Vi v (V2 — > 0) = V2 v = Vi falsifies the law of excluded middle. • Every topology provides a complete Heyting algebra in the form of its open set lattice. In this case, the element A — > B is the interior of the union of A and B, where A denotes the complement of the open set A. Not all complete Heyting algebras are of this form. These issues are studied in pointless topology, where complete Heyting algebras are also called frames or locales. • The Lindenbaum algebra of propositional intuitionistic logic is a Heyting algebra. • The global elements of the subobject classifier f^of an elementary topos form a Heyting algebra; it is the Heyting algebra of truth values of the intuitionistic higher-order logic induced by the topos. The free Heyting algebra over one generator (aka Rieger— Nishimura lattice) Properties General properties The ordering < on a Heyting algebra H can be recovered from the operation — > as follows: for any elements a, b of H, a < b if and only if" <a — > Z? = 1 . In contrast to some many-valued logics, Heyting algebras share the following property with Boolean algebras: if negation has a fixed point (i.e. ->a = a for some a), then the Heyting algebra is the trivial one-element Heyting algebra. Provable identities Given a formula F(A , A ,..., A ) of propositional calculus (using, in addition to the variables, the connectives a, v, -1, — >, and the constants and 1), it is a fact, proved early on in any study of Heyting algebras, that the following two conditions are equivalent: 1 . The formula F is provably true in intuitionist propositional calculus. 2. The identity F(a , a , ..., a )= 1 is true for any Heyting algebra Hand any elements a , a ,..., a EH. The implication 1 — > 2 is extremely useful and is the principal practical method for proving identities in Heyting algebras. In practice, one frequently uses the deduction theorem in such proofs. Since for any a and b in a Heyting algebra H we have a < b if and only if a — > b - 1, it follows from 1 — > 2 that whenever a formula a formula F — > G is provably true, we have F(a , a ,..., a ) < G(a , a ,..., a ) for any Heyting algebra H, and any elements a , a ,..., a € H. (It follows from the deduction theorem that F — » G is provable [from nothing] if and only if G is a provable from F, that is, if G is a provable consequence of F.) In particular, if F and G Hey ting algebra 53 are provably equivalent, then F(a , a ,..., a ) = G(a , a ,..., a ), since < is an order relation. 1 — > 2 can be proved by examining the logical axioms of the system of proof and verifying that their value is 1 in any Heyting algebra, and then verifying that the application of the rules of inference to expressions with value 1 in a Heyting algebra results in expressions with value 1 . For example, let us choose the system of proof having modus ponens as its sole rule of inference, and whose axioms are the Hilbert-style ones given at Intuitionistic logic#Axiomatization. Then the facts to be verified follow immediately from the axiom-like definition of Heyting algebras given above. 1 — > 2 also provides a method for proving that certain propositional formulas, though tautologies in classical logic, cannot be proved in intuitionist propositional logic. In order to prove that some formula F(A , A ,..., A ) is not provable, it is enough to exhibit a Heyting algebra //and elements a,, a.,..., a € //such that F{a,, a^,...,a ) * 1. r o j o o 1 2 « 1 2 n If one wishes to avoid mention of logic, then in practice it becomes necessary to prove as a lemma a version of the deduction theorem valid for Heyting algebras: for any elements a, b and c of a Heyting algebra //, we have {a a b) — > c = a — > (b — » c). For more on the implication 2 — > 1, see the section "Universal constructions" below. Distributivity Heyting algebras are always distributive. Specifically, we always have the identities 1. a A (& V c) = (a A 6) V (a A c) 2. a V (& A c) = (a V 6) A (a V c) The distributive law is sometimes stated as an axiom, but in fact it follows from the existence of relative pseudo-complements. The reason is that, being the lower adjoint of a Galois connection, A preserves all existing suprema. Distributivity in turn is just the preservation of binary suprema by a. By a similar argument, the following infinite distributive law holds in any complete Heyting algebra: xA\jY = \J{xAy.y£Y} for any element x in H and any subset Y of H. Conversely, any complete lattice satisfying the above infinite distributive law is a complete Heyting algebra, with a -^ b = \/{c j a A c < b} being its relative pseudo-complement operation. Regular and complemented elements An element x of a Heyting algebra H is called regular if either of the following equivalent conditions hold: 1. X = -i-ix. 2. x = -<y for some y £ H. The equivalence of these conditions can be restated simply as the identity — 1— 1— uc = ->x, valid for all x&H. Elements x and y of a Heyting algebra H are called complements to each other if x a y = and x v y = 1 . If it exists, any such y is unique and must in fact be equal to ->x. We call an element x complemented if it admits a complement. It is true that if x is complemented, then so is ->x, and then x and -uc are complements to each other. However, confusingly, even if x is not complemented, ->x may nonetheless have a complement (not equal to x). In any Heyting algebra, the elements and 1 are complements to each other. For instance, it is possible that -uc is for every x different from 0, and 1 if x = 0, in which case and 1 are the only regular elements. Any complemented element of a Heyting algebra is regular, though the converse is not true in general. In particular, and 1 are always regular. For any Heyting algebra //, the following conditions are equivalent: Heyting algebra 54 1 . H is a Boolean algebra; 2. every x in H is regular; 3. every x in H is complemented. In this case, the element a — > Zj is equal to ->a v b. The regular (resp. complemented) elements of any Heyting algebra H constitute a Boolean algebra H (resp. reg H ), in which the operations a, -1 and — >, as well as the constants and 1, coincide with those of H. In the case of comp r H , the operation v is also the same, hence H is a subalgebra of H. In general however, H will not be a comp comp reg subalgebra of//, because its join operation v may be differ from v. For x, y € H , we have x v y = — ■(— uc a -> y). f j ± re g .> ^ re g re g ^ See below for necessary and sufficient conditions in order for v to coincide with v. reg The De Morgan laws in a Heyting algebra One of the two De Morgan laws is satisfied in every Heyting algebra, namely ->(x V y) — -ix A -iy, for all x, y £ H . However, the other De Morgan law does not always hold. We have instead a weak de Morgan law: -i(x A y) — -1-1 (-ix V -iy), for all x, y £ if . The following statements are equivalent for all Heyting algebras //: 1 . H satisfies both De Morgan laws, 2. -i(x Ay) — -ix V -iy for all x,y G H, 3. -i(x A y) = -ix V -iy for all regular x,y G H, 4. -i-i(x V y) = -i-ix V — 1 — 13/ for all x,y £ H, 5. -1-1 (x V y) = x V y for all regular x, y € H, 6. -i(-ix A -iy) = x V y for all regular x,y G H, 7. -,x V -.-.x = 1 for all x € H . Condition 2 is the other De Morgan law. Condition 6 says that the join operation v on the Boolean algebra H of regular elements of H coincides with the operation v of H. Condition 7 states that every regular element is complemented, i.e., H = H reg comp We prove the equivalence. Clearly 1 — > 2, 2 — > 3 and 4 — > 5 are trivial. Furthermore, 3 <-> 4 and 5 <-> 6 result simply from the first De Morgan law and the definition of regular elements. We show that 6 — > 7 by taking -w and -i-uc in place of x and y in 6 and using the identity a a -ia = 0. Notice that 2 — > 1 follows from the first De Morgan law, and 7 — > 6 results from the fact that the join operation v on the subalgebra H is just the restriction of v to H , comp comp taking into account the characterizations we have given of conditions 6 and 7. The implication 5 — » 2 is a trivial consequence of the weak De Morgan law, taking -uc and -ry in place of x and y in 5. Heyting algebras satisfying the above properties are related to De Morgan logic in the same way Heyting algebras in general are related to intuitionist logic. Heyting algebra 55 Heyting algebra morphisms Definition Given two Heyting algebras H and H and a mapping/: H — > H , we say that/ is a morphism of Heyting algebras if, for any elements x and y in H , we have: 1. /(0) = 0, 2- /(l) = 1, 3- f(xAy)=f(x)Af(y), 4. f(xVy)=f(x)Vf(y), 5. f{x^y)=f{x)^f{y), 6- /(-*) = -n/(x). We put condition 6 in brackets because it follows from the others, as -uc is just x— >0, and one may or may not wish to consider -1 to be a basic operation. It follows from conditions 3 and 5 (or 1 alone, or 2 alone) that / is an increasing function, that is, that fix) < fly) whenever x < y. Assume H and H are structures with operations — >, a, v (and possibly -1) and constants and 1, and/is a surjective mapping from H to H with properties 1 through 5 (or 1 through 6) above. Then if H is a Heyting algebra, so too is H . This follows from the characterization of Heyting algebras as bounded lattices (thought of as algebraic structures rather than partially ordered sets) with an operation — > satisfying certain identities. Properties The identity map fix) = x from any Heyting algebra to itself is a morphism, and the composite g u f of any two morphisms/and g is a morphism. Hence Heyting algebras form a category. Examples Given a Heyting algebra H and any subalgebra H , the inclusion mapping i : H — » H is a morphism. For any Heyting algebra H, the map x h- > — 1— uc defines a morphism from H onto the Boolean algebra of its regular elements H . This is not in general a morphism from H to itself, since the join operation of H may be different from that of H. Quotients Let H be a Heyting algebra, and let F Q H. We call F a filter on H if it satisfies the following properties: 1. leF, 2. ]fx,y £ F then x Ay £ F, 3. If x £ .F, y G H, and x < y then y £ F. The intersection of any set of filters on // is again a filter. Therefore, given any subset 5 of H there is a smallest filter containing S. We call it the filter generated by S. If S is empty, F - { 1 } . Otherwise, F is equal to the set of x in H such that there exist v., y„, ..., y € S 1 with y, a y„ a ... a y < x. 12 n 12 n If // is a Heyting algebra and F is a filter on //, we define a relation ~ on H as follows: we write x ~ y whenever x — » y and y — > x both belong to F. Then ~ is an equivalence relation; we write HIF for the quotient set. There is a unique Heyting algebra structure on HIF such that the canonical surjection p : H — > HIF becomes a Heyting algebra morphism. We call the Heyting algebra HIF the quotient of H by F. Let S be a subset of a Heyting algebra H and let F be the filter generated by S. Then HIF satisfies the following universal property: Heyting algebra 56 • Given any morphism of Heyting algebras/: H — > //' satisfying /(jy) = 1 for every y £ S, /factors uniquely through the canonical surjection p : H — > HIF. That is, there is a unique morphism/' : HIF — > //' satisfying/'/? =/. The morphism/' is said to be induced by/ Let/: H — » // be a morphism of Heyting algebras. The kernel of/ written ker/ is the set/ - [{ 1 }]. It is a filter on H . (Care should be taken because this definition, if applied to a morphism of Boolean algebras, is dual to what would be called the kernel of the morphism viewed as a morphism of rings.) By the foregoing, /induces a morphism /' : H /(ker/) — » // . It is an isomorphism of H /(ker/) onto the subalgebra/// ] of H . Universal constructions Heyting algebra of propositional formulas in n variables up to intuitionist equivalence The implication 2— >1 in the section "Provable identities" is proved by showing that the result of the following construction is itself a Heyting algebra: 1. Consider the set L of propositional formulas in the variables A , A ,..., A . 2. Endow L with a preorder < by defining F<G if G is an (intuitionist) logical consequence of F, that is, if G is provable from F. It is immediate that < is a preorder. 3. Consider the equivalence relation F~G induced by the preorder F<G. (It is defined by F~G if and only if F<G and G<F. In fact, ~ is the relation of (intuitionist) logical equivalence.) 4. Let H be the quotient set Ll~. This will be the desired Heyting algebra. 5. We write [F] for the equivalence class of a formula F. Operations — >, a, v and -> are defined in an obvious way on L. Verify that given formulas F and G, the equivalence classes [F— >G], [FaG], [FvG] and [->F] depend only on [F] and [G]. This defines operations — >, a, v and -> on the quotient set H =LI~. Further define 1 to be the class of provably true statements, and set 0=[-L]. 6. Verify that H , together with these operations, is a Heyting algebra. We do this using the axiom-like definition of Heyting algebras. H satisfies conditions THEN-1 through FALSE because all formulas of the given forms are axioms of intuitionist logic. MODUS -PONENS follows from the fact that if a formula T— >/*" is provably true, where T is provably true, then F is provably true (by application of the rule of inference modus ponens). Finally, EQUIV results from the fact that if F— >G and G^>F are both provably true, then F and G are provable from each other (by application of the rule of inference modus ponens), hence [F]=[G]. As always under the axiom-like definition of Heyting algebras, we define < on H by the condition that x<y if and only if x— >;y=l. Since, by the deduction theorem, a formula F^>G is provably true if and only if G is provable from F, it follows that [F]<[G] if and only if F<G. In other words, < is the order relation on Ll~ induced by the preorder < onL. Free Heyting algebra on an arbitrary set of generators In fact, the preceding construction can be carried out for any set of variables {A.: /€/} (possibly infinite). One obtains in this way the free Heyting algebra on the variables {A.}, which we will again denote by H . It is free in the sense that given any Heyting algebra H given together with a family of its elements (a.: /€/ ) , there is a unique morphism f.H—>H satisfying f([A])=a.. The uniqueness of / is not difficult to see, and its existence results essentially from the implication 1— >2 of the section "Provable identities" above, in the form of its corollary that whenever F and G are provably equivalent formulas, F( (a.) )=G( (a) ) for any family of elements (a.) in//. Heyting algebra 57 Heyting algebra of formulas equivalent with respect to a theory T Given a set of formulas Tin the variables {A.}, viewed as axioms, the same construction could have been carried out with respect to a relation F<G defined on L to mean that G is a provable consequence of F and the set of axioms T. Let us denote by H the Heyting algebra so obtained. Then H satisfies the same universal property as H above, but with respect to Heyting algebras H and families of elements (a.) satisfying the property that /( (a) )=1 for any axiom /( (A.) ) in T. (Let us note that H r taken with the family of its elements ([A.]) , itself satisfies this property.) The existence and uniqueness of the morphism is proved the same way as for H , except that one must modify the implication 1— >2 in "Provable identities" so that 1 reads "provably true from T," and 2 reads "any elements a , a ,..., a in H satisfying the formulas ofT." The Heyting algebra H that we have just defined can be viewed as a quotient of the free Heyting algebra H on the same set of variables, by applying the universal property of H with respect to H r and the family of its elements <[A.]> . 1 Every Heyting algebra is isomorphic to one of the form H . To see this, let H be any Heyting algebra, and let (a;. /€l) be a family of elements generating H (for example, any surjective family). Now consider the set T of formulas /( (A.) ) in the variables (A.: 2GI) such that /( (a.) )=1. Then we obtain a morphism/:// —>// by the universal property of H r which is clearly surjective. It is not difficult to show that/is injective. Comparison to Lindenbaum algebras The constructions we have just given play an entirely analogous role with respect to Heyting algebras to that of Lindenbaum algebras with respect to Boolean algebras. In fact, The Lindenbaum algebra B in the variables {A.} with respect to the axioms T is just our H 1, where T is the set of all formulas of the form -r-iF-^F, since the additional axioms of T are the only ones that need to be added in order to make all classical tautologies provable. Heyting algebras as applied to intuitionistic logic If one interprets the axioms of the intuitionistic propositional logic as terms of a Heyting algebra, then they will evaluate to the largest element, 1, in any Heyting algebra under any assignment of values to the formula's variables. For instance, (P A Q) — > .Pis, by definition of the pseudo-complement, the largest element x such that P A Q A x < P . This inequation is satisfied for any x, so the largest such x is 1. Furthermore the rule of modus ponens allows us to derive the formula Q from the formulas P and P — > Q. But in any Heyting algebra, if P has the value 1, and P — > Q has the value 1, then it means that P A 1 < Q , and so 1 A 1 < Q ; it can only be that Q has the value 1 . This means that if a formula is deducible from the laws of intuitionistic logic, being derived from its axioms by way of the rule of modus ponens, then it will always have the value 1 in all Heyting algebras under any assignment of values to the formula's variables. However one can construct a Heyting algebra in which the value of Peirce's law is not always 1. Consider the 3-element algebra {(V/2,1 } as given above. If we assign Vi to P and to Q, then the value of Peirce's law ((P — > Q) — > P) — > P is Vi. It follows that Peirce's law cannot be intuitionistically derived. See Curry— Howard isomorphism for the general context of what this implies in type theory. The converse can be proven as well: if a formula always has the value 1, then it is deducible from the laws of intuitionistic logic, so the intuitionistically valid formulas are exactly those that always have a value of 1. This is similar to the notion that classically valid formulas are those formulas that have a value of 1 in the two-element Boolean algebra under any possible assignment of true and false to the formula's variables — that is, they are formulas which are tautologies in the usual truth-table sense. A Heyting algebra, from the logical standpoint, is then a generalization of the usual system of truth values, and its largest element 1 is analogous to 'true'. The usual two-valued logic system is a special case of a Heyting algebra, and the smallest non-trivial one, in which the only elements of the algebra are 1 (true) and (false). Hey ting algebra 58 Word problem T31 The word problem on free Hey ting algebras is difficult. The only known results are that the free Hey ting algebra on one generator is infinite, and that the free complete Heyting algebra on one generator exists (and has one more element than the free Heyting algebra). Notes [1] Rutherford (1965), Th.26.2 p.78. [2] Rutherford (1965), Th.26.1 p.78. [3] Peter T. Johnstone, Stone Spaces, (1982) Cambridge University Press, Cambridge, ISBN 0-521-23893-5. (See paragraph 4.11) References • Rutherford, Daniel Edwin (1965). Introduction to Lattice Theory. Oliver and Boyd. • F. Borceux, Handbook of Categorical Algebra 3, In Encyclopedia of Mathematics and its Applications, Vol. 53, Cambridge University Press, 1994. • G. Gierz, K.H. Hoffmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott, Continuous Lattices and Domains, In Encyclopedia of Mathematics and its Applications, Vol. 93, Cambridge University Press, 2003. • S. Ghilardi. Free Heyting algebras as bi-Heyting algebras, Math. Rep. Acad. Sci. Canada XVI., 6:240—244, 1992. External links • Heyting algebra (GFDLed) MV-algebra In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation © , a unary operation — i, and the constant Q, satisfying certain axioms. MV-algebras are models of Lukasiewicz logic; the letters MV refer to multi-valued logic of Lukasiewicz. Definitions An MV-algebra is an algebraic structure /A, ©, — i, 0) , consisting of • a non-empty set A , • a binary operation © on A , • a unary operation — i on A , and • a constant Q denoting a fixed element of A, which satisfies the following identities: • {x®y)®z = x®{y@z), • x © = x, • x®y — y®x : • -i-ii = x, • X© -iO = -iO,and • ->(->x@y)®y = ->(-y®x)®x. By virtue of the first three axioms, I A, ©, 0) is a commutative monoid. Being defined by identities, MV-algebras form a variety of algebras. The variety of MV-algebras is a subvariety of the variety of BL-algebras and contains all Boolean algebras. MV-algebra 59 An MV-algebra can equivalently be defined (Hajek 1998) as a prelinear commutative bounded integral residuated lattice (£,, A, V, ®, — ►, 0, 1} satisfying the additional identity x V y — {x — > y) — ► y. Examples of MV-algebras A simple numerical example is A = [0, ll,with operations x © y = min(:r + y, l)and —,x = 1 — x. in mathematical fuzzy logic, this MV-algebra is called the standard MV-algebra, as it forms the standard real-valued semantics of Lukasiewicz logic. The trivial MV-algebra has the only element and the operations defined in the only possible way, © = an d ^0 = 0. The two-element MV-algebra is actually the two-element Boolean algebra {0, 1} with © coinciding with Boolean disjunction and —1 with Boolean negation. Other finite linearly ordered MV-algebras are obtained by restricting the universe and operations of the standard MV-algebra to the set of 77, -\- 1 equidistant real numbers between and 1 (both included), that is, the set {0, 1/n, 2/n, . . . , 1}, which is closed under the operations © and —1 of the standard MV-algebra. Another important example is Chang's MV-algebra, consisting just of infinitesimals (with the order type a>) and their co-infinitesimals. Relation to Lukasiewicz logic Chang devised MV-algebras to study multi-valued logics, introduced by Jan Lukasiewicz in 1920. In particular, MV-algebras form the algebraic semantics of Lukasiewicz logic, as described below. Given an MV-algebra A, an A-valuation is a homomorphism from the algebra of propositional formulas (in the language consisting of ©, — i,and 0) into A. Formulas mapped to 1 (or — 10) for all A-valuations are called A-tautologies. If the standard MV-algebra over [0,1] is employed, the set of all [0,1] -tautologies determines so-called infinite-valued Lukasiewicz logic. Chang's (1958, 1959) completeness theorem states that any MV-algebra equation holding in the standard MV-algebra over the interval [0,1] will hold in every MV-algebra. Algebraically, this means that the standard MV-algebra generates the variety of all MV-algebras. Equivalently, Chang's completeness theorem says that MV-algebras characterize infinite-valued Lukasiewicz logic, defined as the set of [0,l]-tautologies. The way the [0,1] MV-algebra characterizes all possible MV-algebras parallels the well-known fact that identities holding in the two-element Boolean algebra hold in all possible Boolean algebras. Moreover, MV-algebras characterize infinite-valued Lukasiewicz logic in a manner analogous to the way that Boolean algebras characterize classical bivalent logic (see Lindenbaum-Tarski algebra). References • Chang, C. C. (1958) "Algebraic analysis of many-valued logics," Transactions of the American Mathematical Society 88: 476-490. • (1959) "A new proof of the completeness of the Lukasiewicz axioms," Transactions of the American Mathematical Society 88: 74—80. • Cignoli, R. L. O., D'Ottaviano, I. M. L., Mundici, D. (2000) Algebraic Foundations of Many-valued Reasoning. Kluwer. • Di Nola A., Lettieri A. (1993) "Equational characterization of all varieties of MV-algebras," Journal of Algebra 221: 123-131. • Hajek, Petr (1998) Metamathematics of Fuzzy Logic. Kluwer. MV-algebra 60 External links Hi • Stanford Encyclopedia of Philosophy: "Many-valued logic — by Siegfried Gottwald. Group theory In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have strongly influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced tremendous advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, can be modelled by symmetry groups. Thus group theory and the closely related representation theory have many applications in physics and chemistry. One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 1980, that culminated in a complete classification of finite simple groups. History Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields. Early results about permutation groups were obtained by Lagrange, Ruffini, and Abel in their quest for general solutions of polynomial equations of high degree. Evariste Galois coined the term "group" and established a connection, now known as Galois theory, between the nascent theory of groups and field theory. In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. Felix Klein's Erlangen program famously proclaimed group theory to be the organizing principle of geometry. Galois, in the 1830s, was the first to employ groups to determine the solvability of polynomial equations. Arthur Cayley and Augustin Louis Cauchy pushed these investigations further by creating the theory of permutation group. The second historical source for groups stems from geometrical situations. In an attempt to come to grips with possible geometries (such as euclidean, hyperbolic or projective geometry) using group theory, Felix Klein initiated the Erlangen programme. Sophus Lie, in 1884, started using groups (now called Lie groups) attached to analytic problems. Thirdly, groups were (first implicitly and later explicitly) used in algebraic number theory. The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since then, the impact of group theory has been ever growing, giving rise to the birth of abstract algebra in the early 20th century, representation theory, and many more influential spin-off domains. The classification of finite simple groups is a vast body of work from the mid 20th century, classifying all the finite simple groups. Group theory 61 Main classes of groups The range of groups being considered has gradually expanded from finite permutation groups and special examples of matrix groups to abstract groups that may be specified through a presentation by generators and relations. Permutation groups The first class of groups to undergo a systematic study was permutation groups. Given any set X and a collection G of bijections of X into itself (known as permutations) that is closed under compositions and inverses, G is a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group S ; in general, G is a subgroup of the symmetric group of X. An early construction due to Cayley exhibited any group as a permutation group, acting on itself (X = G) by means of the left regular representation. In many cases, the structure of a permutation group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n > 5, the alternating group A is simple, i.e. does not admit any proper normal subgroups. This fact plays a key role in the impossibility of solving a general algebraic equation of degree n > 5 in radicals. Matrix groups The next important class of groups is given by matrix groups, or linear groups. Here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such a group acts on the «-dimensional vector space K" by linear transformations. This action makes matrix groups conceptually similar to permutation groups, and geometry of the action may be usefully exploited to establish properties of the group G. Transformation groups Permutation groups and matrix groups are special cases of transformation groups: groups that act on a certain space X preserving its inherent structure. In the case of permutation groups, X is a set; for matrix groups, X is a vector space. The concept of a transformation group is closely related with the concept of a symmetry group: transformation groups frequently consist of all transformations that preserve a certain structure. The theory of transformation groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, considers group actions on manifolds by homeomorphisms or diffeomorphisms. The groups themselves may be discrete or continuous. Abstract groups Most groups considered in the first stage of the development of group theory were "concrete", having been realized through numbers, permutations, or matrices. It was not until the late nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by generators and relations, G= (S\R). A significant source of abstract groups is given by the construction of a factor group, or quotient group, GIH, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory. If a group G is a permutation group on a set X, the factor group GIH is no longer acting on X; but the idea of an abstract group permits one not to worry about this discrepancy. The change of perspective from concrete to abstract groups makes it natural to consider properties of groups that are independent of a particular realization, or in modern language, invariant under isomorphism, as well as the classes of group with a given such property: finite groups, periodic groups, simple groups, solvable groups, and so on. Rather than exploring properties of an individual group, one seeks to establish results that apply to a whole class of groups. Group theory 62 The new paradigm was of paramount importance for the development of mathematics: it foreshadowed the creation of abstract algebra in the works of Hilbert, Emil Artin, Emmy Noether, and mathematicians of their school. Topological and algebraic groups An important elaboration of the concept of a group occurs if G is endowed with additional structure, notably, of a topological space, differentiable manifold, or algebraic variety. If the group operations m (multiplication) and i (inversion), m : G X G — > G, (g : h) i— > gh : i : G — > G,g i— > <? _1 , are compatible with this structure, i.e. are continuous, smooth or regular (in the sense of algebraic geometry) maps then G becomes a topological group, a Lie group, or an algebraic group. The presence of extra structure relates these types of groups with other mathematical disciplines and means that more tools are available in their study. Topological groups form a natural domain for abstract harmonic analysis, whereas Lie groups (frequently realized as transformation groups) are the mainstays of differential geometry and unitary representation theory. Certain classification questions that cannot be solved in general can be approached and resolved for special subclasses of groups. Thus, compact connected Lie groups have been completely classified. There is a fruitful relation between infinite abstract groups and topological groups: whenever a group F can be realized as a lattice in a topological group G, the geometry and analysis pertaining to G yield important results about F. A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups (profinite groups): for example, a single p-adic analytic group G has a family of quotients which are finite p-groups of various orders, and properties of G translate into the properties of its finite quotients. Combinatorial and geometric group theory Groups can be described in different ways. Finite groups can be described by writing down the group table consisting of all possible multiplications g • h. A more important way of defining a group is by generators and relations, also called the presentation of a group. Given any set F of generators {g.} r the free group generated by F surjects onto the group G. The kernel of this map is called subgroup of relations, generated by some subset D. The presentation is usually denoted by (F I D ) . For example, the group Z = (a I ) can be generated by one element a (equal to +1 or -1) and no relations, because «1 never equals unless n is zero. A string consisting of generator symbols is called a word. Combinatorial group theory studies groups from the perspective of generators and relations. It is particularly useful where finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite). The area makes use of the connection of graphs via their fundamental groups. For example, one can show that every subgroup of a free group is free. There are several natural questions arising from giving a group by its presentation. The word problem asks whether two words are effectively the same group element. By relating the problem to Turing machines, one can show that there is in general no algorithm solving this task. An equally difficult problem is, whether two groups given by different presentations are actually isomorphic. For example Z can also be presented by (x, y I xyxyx = 1) and it is not obvious (but true) that this presentation is isomorphic to the standard one above. Group theory 63 Geometric group theory attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on. The first idea is made precise by means of the Cayley graph, whose vertices correspond to group elements and edges correspond to right multiplication in the group. Given two elements, one constructs the word metric given by the length of the minimal path between the elements. A theorem of Milnor and Svarc then says that given a group G acting in a reasonable manner on a metric space X, for example a compact manifold, then G is quasi-isometric (i.e. looks similar from the far) to the space X. ^+ + ** 4 +.$, r Jt-* P € .*k ** <$- — 4; 4 .+.? P P P. P P The Cayley graph of ( x, y I }, the free group of rank 2. Representation of groups Saying that a group G acts on a set X means that every element defines a bijective map on a set in a way compatible with the group structure. When X has more structure, it is useful to restrict this notion further: a representation of G on a vector space V is a group homomorphism: q : G -> GL(V), where GL(V) consists of the invertible linear transformations of V. In other words, to every group element g is assigned an automorphism g(g) such that g(g) g{h) = g{gh) for any h in G. Ml This definition can be understood in two directions, both of which give rise to whole new domains of mathematics. On the one hand, it may yield new information about the group G: often, the group operation in G is abstractly given, but via q, it corresponds to the multiplication of matrices, which is very explicit. On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if G is finite, it is known that V above decomposes into irreducible parts. These parts in turn are much more easily manageable than the whole V (via Schur's lemma). Given a group G, representation theory then asks what representations of G exist. There are several settings, and the employed methods and obtained results are rather different in every case: representation theory of finite groups and representations of Lie groups are two main subdomains of the theory. The totality of representations is governed by the group's characters. For example, Fourier polynomials can be interpreted as the characters of U(l), the group of 2 complex numbers of absolute value 7, acting on the L -space of periodic functions. Connection of groups and symmetry Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example 1. If X is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups. 2. If the object X is a set of points in the plane with its metric structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (an isometry). The corresponding group is called isometry group of X. 3. If instead angles are preserved, one speaks of conformal maps. Conformal maps give rise to Kleinian groups, for example. 4. Symmetries are not restricted to geometrical objects, but include algebraic objects as well. For instance, the equation Group theory 64 x 2 - 3 = has the two solutions _i_«/g, and _v/3- in m i s case > the group that exchanges the two roots is the Galois group belonging to the equation. Every polynomial equation in one variable has a Galois group, that is a certain permutation group on its roots. The axioms of a group formalize the essential aspects of symmetry. Symmetries form a group: they are closed because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry. The identity keeping the object fixed is always a symmetry of an object. Existence of inverses is guaranteed by undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions are associative. Frucht's theorem says that every group is the symmetry group of some graph. So every abstract group is actually the symmetries of some explicit object. The saying of "preserving the structure" of an object can be made precise by working in a category. Maps preserving the structure are then the morphisms, and the symmetry group is the automorphism group of the object in question. Applications of group theory Applications of group theory abound. Almost all structures in abstract algebra are special cases of groups. Rings, for example, can be viewed as abelian groups (corresponding to addition) together with a second operation (corresponding to multiplication). Therefore group theoretic arguments underlie large parts of the theory of those entities. Galois theory uses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). The fundamental theorem of Galois theory provides a link between algebraic field extensions and group theory. It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding Galois group. For example, S , the symmetric group in 5 elements, is not solvable which implies that the general quintic equation cannot be solved by radicals in the way equations of lower degree can. The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as class field theory. Algebraic topology is another domain which prominently associates groups to the objects the theory is interested in. There, groups are used to describe certain invariants of topological spaces. They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some deformation. For example, the fundamental group "counts" how many paths in the space are essentially different. The Poincare conjecture, proved in 2002/2003 by Grigori Perelman is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use of Eilenberg— MacLane spaces which are spaces with prescribed homotopy groups. Similarly algebraic K-theory stakes in a crucial way on classifying spaces of groups. Finally, the name of the torsion subgroup of an infinite group shows the legacy of topology in group theory. Algebraic geometry and cryptography likewise uses group theory in many ways. Abelian varieties have been introduced above. The presence of the group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures. The one-dimensional case, namely elliptic curves is studied in particular detail. They are both theoretically and practically intriguing. Very large groups of prime order constructed in Elliptic -Curve Cryptography serve for public key cryptography. Cryptographical methods of this kind A toms Its abdian group stmcture is induced from the map C — > C/Z+rZ, where r is a parameter. Group theory 65 iXi YiZ A B C D E F ^^ .*....*.....*.. A B | C | D | E | F GH 1 The cyclic group Z underlies Caesar's cipher. benefit from the flexibility of the geometric objects, hence their group structures, together with the complicated structure of these groups, which make the discrete logarithm very hard to calculate. One of the earliest encryption protocols, Caesar's cipher, may also be interpreted as a (very easy) group operation. In another direction, toric varieties are algebraic varieties acted on by a torus. Toroidal embeddings have recently led to advances in ro] algebraic geometry, in particular resolution of singularities. Algebraic number theory is a special case of group theory, thereby following the rules of the latter. For example, Euler's product formula 1 Hn p prime captures the fact that any integer decomposes in a unique way into primes. The failure of this statement for more general rings gives rise to class groups and regular primes, which feature in Kummer's treatment of Fermat's Last Theorem. • The concept of the Lie group (named after mathematician Sophus Lie) is important in the study of differential equations and manifolds; they describe the symmetries of continuous geometric and analytical structures. Analysis on these and other groups is called harmonic analysis. Haar measures, that is integrals invariant under the translation in a Lie group, are used for pattern recognition and other image processing techniques. • In combinatorics, the notion of permutation group and the concept of group action are often used to simplify the counting of a set of objects; see in particular Burnside's lemma. • The presence of the 12-periodicity in the circle of fifths yields applications of elementary group theory in musical set theory. • In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories. Examples of the use of groups in physics include the Standard Model, gauge theory, the Lorentz group, and the Poincare group. • In chemistry and materials science, groups are used to classify crystal structures, regular polyhedra, and the symmetries of molecules. The assigned point groups can then be used to determine physical properties (such as polarity and chirality), spectroscopic properties (particularly useful for Raman spectroscopy and infrared spectroscopy), and to construct molecular orbitals. jT 1 c G Bb/f 1 V EbN 3] A AbV 6 4/ */ E (C#) (Gb) B The circle of fifths may be endowed with a cyclic group structure Group theory 66 See also • Group (mathematics) • Glossary of group theory • List of group theory topics Notes [1] This process of imposing extra structure has been formalized through the notion of a group object in a suitable category. Thus Lie groups are group objects in the category of differentiable manifolds and affine algebraic groups are group objects in the category of affine algebraic varieties. [2] Schupp & Lyndon 2001 [3] LaHarpe2000 [4] Such as group cohomology or equivariant K-theory. [5] In particular, if the representation is faithful. [6] For example the Hodge conjecture (in certain cases). [7] See the Birch-S winnerton-Dyer conjecture, one of the millennium problems [8] Abramovich, Dan; Karu, Kalle; Matsuki, Kenji; Wlodarczyk, Jaroslaw (2002), "Torification and factorization of birational maps", Journal of the American Mathematical Society 15 (3): 531-572, doi:10.1090/S0894-0347-02-00396-X, MR1896232 [9] Lenz, Reiner (1990), Group theoretical methods in image processing (http://webstaff.itn.liu.se/~reile/LNCS413/index.htm), Lecture Notes in Computer Science, 413, Berlin, New York: Springer- Verlag, doi:10.1007/3-540-52290-5, ISBN 978-0-387-52290-6, References • Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics, 126 (2nd ed.), Berlin, New York: Springer- Verlag, MR1 102012, ISBN 978-0-387-97370-8 • Carter, Nathan C. (2009), Visual group theory (http://web.bentley.edu/empl/c/ncarter/vgt/), Classroom Resource Materials Series, Mathematical Association of America, MR2504193, ISBN 978-0-88385-757-1 • Cannon, John J. (1969), "Computers in group theory: A survey", Communications of the Association for Computing Machinery 12: 3-12, doi: 10. 1145/362835.362837, MR0290613 • Frucht, R. (1939), "Herstellung von Graphen mit vorgegebener abstrakter Gruppe" (http://www.numdam.org/ numdam-bin/fitem?id=CM_1939_6_239_0), Compositio Mathematica 6: 239-50, ISSN 0010-437X • Golubitsky, Martin; Stewart, Ian (2006), "Nonlinear dynamics of networks: the groupoid formalism", Bull. Amer. Math. Soc. (N.S.) 43: 305-364, doi: 10. 1090/S0273-0979-06-01 108-6, MR2223010 Shows the advantage of generalising from group to groupoid. • Judson, Thomas W. (1997), Abstract Algebra: Theory and Applications (http://abstract.ups.edu) An introductory undergraduate text in the spirit of texts by Gallian or Herstein, covering groups, rings, integral domains, fields and Galois theory. Free downloadable PDF with open-source GFDL license. • Kleiner, Israel (1986), "The evolution of group theory: a brief survey" (http://jstor.org/stable/2690312), Mathematics Magazine 59 (4): 195-215, doi:10.2307/2690312, MR863090, ISSN 0025-570X • La Harpe, Pierre de (2000), Topics in geometric group theory, University of Chicago Press, ISBN 978-0-226-31721-2 • Livio, M. (2005), The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry, Simon & Schuster, ISBN 0-7432-5820-7 Conveys the practical value of group theory by explaining how it points to symmetries in physics and other sciences. • Mumford, David (1970), Abelian varieties, Oxford University Press, ISBN 978-0-19-560528-0, OCLC 138290 • Ronan M., 2006. Symmetry and the Monster. Oxford University Press. ISBN 0-19-280722-6. For lay readers. Describes the quest to find the basic building blocks for finite groups. • Rotman, Joseph (1994), An introduction to the theory of groups, New York: Springer- Verlag, ISBN 0-387-94285-8 A standard contemporary reference. Group theory 67 • Schupp, Paul E.; Lyndon, Roger C. (2001), Combinatorial group theory, Berlin, New York: Springer- Verlag, ISBN 978-3-540-41 158-1 • Scott, W. R. (1987) [1964], Group Theory, New York: Dover, ISBN 0-486-65377-3 Inexpensive and fairly readable, but somewhat dated in emphasis, style, and notation. • Shatz, Stephen S. (1972), Profinite groups, arithmetic, and geometry, Princeton University Press, MR0347778, ISBN 978-0-691-08017-8 • Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, MR1269324, ISBN 978-0-521-55987-4, OCLC 36131259 External links • History of the abstract group concept (http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/ Abstract_groups.html) • Higher dimensional group theory (http://www.bangor.ac.Uk/r.brown/hdaweb2.htm) This presents a view of group theory as level one of a theory which extends in all dimensions, and has applications in homotopy theory and to higher dimensional nonabelian methods for local-to-global problems. • Plus teacher and student package: Group Theory (http://plus.maths.org/issue48/package/index.html) This package brings together all the articles on group theory from Plus, the online mathematics magazine produced by the Millennium Mathematics Project at the University of Cambridge, exploring applications and recent breakthroughs, and giving explicit definitions and examples of groups. • US Naval Academy group theory guide (http://www.usna.edu/Users/math/wdj/tonybook/gpthry/nodel. html) A general introduction to group theory with exercises written by Tony Gaglione. Abelian group 68 Abelian group Concepts in group theory category of groups subgroups, normal subgroups group homomorphisms, kernel, image, quotient direct product, direct sum semidirect product, wreath product Types of groups simple, finite, infinite discrete, continuous multiplicative, additive cyclic, abelian, dihedral nilpotent, solvable list of group theory topics glossary of group theory An abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity). Abelian groups generalize the arithmetic of addition of integers. They are named after Niels Henrik Abel. The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, with many other basic objects, such as a module and a vector space, being its refinements. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood. On the other hand, the theory of infinite abelian groups is an area of current research. Definition An abelian group is a set, A, together with an operation "•" that combines any two elements a and b to form another element denoted a • b. The symbol "•" is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, (A, •), must satisfy five requirements known as the abelian group axioms: Closure For all a, b in A, the result of the operation a • b is also in A. Associativity For all a, b and c in A, the equation {a • b) • c = a • (b • c) holds. Identity element There exists an element e in A, such that for all elements a in A, the equation e • a = a' e = a holds. Inverse element For each a in A, there exists an element b in A such that a' b = b • a- e, where e is the identity element. Commutativity For all a, b in A, a • b = b • a. More compactly, an abelian group is a commutative group. A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group". Abelian group 69 Facts Notation There are two main notational conventions for abelian groups — additive and multiplicative. Convention Operation Identity Powers Inverse Addition x + y nx -x Multiplication x*yorxy e or 1 n -1 Generally, the multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules. The additive notation may also be used to emphasize that a particular group is abelian, whenever both abelian and non-abelian groups are considered. Multiplication table To verify that a finite group is abelian, a table (matrix) - known as a Cayley table - can be constructed in a similar fashion to a multiplication table. If the group is G = {g = e, g , ..., g } under the operation •, the (/, j)'th entry of this table contains the product g. • g.. The group is abelian if and only if this table is symmetric about the main diagonal ' J (i.e. if the matrix is a symmetric matrix). This is true since if the group is abelian, then g. ■ g.= g. ■ g.. This implies that the (/, j)'th entry of the table equals the * J J * (/', O'th entry - i.e. the table is symmetric about the main diagonal. Examples • For the integers and the operation addition "+", denoted (Z,+), the operation + combines any two integers to form a third integer, addition is associative, zero is the additive identity, every integer n has an additive inverse, -n, and the addition operation is commutative since m + n = n + m for any two integers m and n. • Every cyclic group G is abelian, because if x, y are in G, then xy = a a = a = a = a a =yx. Thus the integers, Z, form an abelian group under addition, as do the integers modulo n, TJriL. • Every ring is an abelian group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group. In particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication. • Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. In general, matrices, even invertible matrices, do not form an abelian group under multiplication because matrix multiplication is generally not commutative. However, some groups of matrices are abelian groups under matrix multiplication - one example is the group of 2x2 rotation matrices. Abelian group 70 Historical remarks Abelian groups were named for Norwegian mathematician Niels Henrik Abel by Camille Jordan because Abel found that the commutativity of the group of an equation implies its roots are solvable by radicals. See Section 6.5 of Cox (2004) for more information on the historical background. Properties If n is a natural number and x is an element of an abelian group G written additively, then nx can be defined as x + x + ... + x (n summands) and {-n)x = -{nx). In this way, G becomes a module over the ring Z of integers. In fact, the modules over Z can be identified with the abelian groups. Theorems about abelian groups (i.e. modules over the principal ideal domain Z) can often be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classification of finitely generated abelian groups which is a specialization of the structure theorem for finitely generated modules over a principal ideal domain. In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direct sum of a torsion group and a free abelian group. The former may be written as a direct sum of finitely many groups of the form Zip Z for p prime, and the latter is a direct sum of finitely many copies of Z. If/, g : G — > H are two group homomorphisms between abelian groups, then their sum/+ g, defined by (f+ g)(x) = J{x) + g(x), is again a homomorphism. (This is not true if H is a non-abelian group.) The set Hom(G, H) of all group homomorphisms from GtoH thus turns into an abelian group in its own right. Somewhat akin to the dimension of vector spaces, every abelian group has a rank. It is defined as the cardinality of the largest set of linearly independent elements of the group. The integers and the rational numbers have rank one, as well as every subgroup of the rationals. Finite abelian groups Cyclic groups of integers modulo n, Z/nZ, were among the first examples of groups. It turns out that an arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The automorphism group of a finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper of Georg Frobenius and Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of linear algebra. Classification The fundamental theorem of finite abelian groups states that every finite abelian group G can be expressed as the direct sum of cyclic subgroups of prime-power order. This is a special case of the fundamental theorem of finitely generated abelian groups when G has zero rank. The cyclic group Z mn ,of order mn is isomorphic to the direct sum of Z m and Z n if and only if m and n are coprime. It follows that any finite abelian group G is isomorphic to a direct sum of the form z fcl © ■ ■ ■ © z fe „ in either of the following canonical ways: • the numbers k,...,k are powers of primes • k, divides fc„, which divides fc„, and so on up to k . 12 3 L u For example, Zi5can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: Z15 = {0, 5, 10} © {0, 3, 6, 9, 12} • The same can be said for any abelian group of order 15, leading to the remarkable conclusion that all abelian groups of order 15 are isomorphic. Abelian group 7 1 For another example, every abelian group of order 8 is isomorphic to either Zs(the integers to 7 under addition modulo 8), Z4 © ^2( m e odd integers 1 to 15 under multiplication modulo 16), or Z2 ffi Z2 © 7Li- See also list of small groups for finite abelian groups of order 16 or less. Automorphisms One can apply the fundamental theorem to count (and sometimes determine) the automorphisms of a given finite abelian group G. To do this, one uses the fact (which will not be proved here) that if G splits as a direct sum H © K of subgroups of coprime order, then Aut(H © K) c^ Aut(ff) © Aut(K). Given this, the fundamental theorem shows that to compute the automorphism group of G it suffices to compute the automorphism groups of the Sylow p-subgroups separately (that is, all direct sums of cyclic subgroups, each with order a power of/?). Fix a prime p and suppose the exponents e. of the cyclic factors of the Sylow p-subgroup are arranged in increasing order: ei < e 2 < ■ ■ ■ < e n for some n > 0. One needs to find the automorphisms of 1 p *i © ■ • ■ © z pe „ . One special case is when n = 1, so that there is only one cyclic prime-power factor in the Sylow p-subgroup P. In this case the theory of automorphisms of a finite cyclic group can be used. Another special case is when n is arbitrary but e. = 1 for 1 < i < n. Here, one is considering P to be of the form uLip (i? ■ ■ ■ ty ^pi so elements of this subgroup can be viewed as comprising a vector space of dimension n over the finite field of p elements W p . The automorphisms of this subgroup are therefore given by the invertible linear transformations, so Aut(P) = GL(n,F p ), where GL is the appropriate general linear group. This is easily shown to have order \Aut(P)\ = (p n -l)---(p"-p n - 1 ). In the most general case, where the e. and n are arbitrary, the automorphism group is more difficult to determine. It is known, however, that if one defines rfjt = maxlrjej. = e fe } and c fe = miri{r|e r = e k } then one has in particular d >k,c< k, and iAut(p)i= n? 4 -/- 1 (fi^r-A ( n fe*- 1 )"-^ 1 One can check that this yields the orders in the previous examples as special cases (see [Hillar,Rhea]). Abelian group 72 Infinite abelian groups The simplest infinite abelian group is the infinite cyclic group Z. Any finitely generated abelian group A is isomorphic to the direct sum of r copies of Z and a finite abelian group, which in turn is decomposable into a direct sum of finitely many cyclic groups of primary orders. Even though the decomposition is not unique, the number r, called the rank of A, and the prime powers giving the orders of finite cyclic summands are uniquely determined. By contrast, classification of general infinitely generated abelian groups is far from complete. Divisible groups, i.e. abelian groups A in which the equation nx = a admits a solution x G A for any natural number n and element a of A, constitute one important class of infinite abelian groups that can be completely characterized. Every divisible group is isomorphic to a direct sum, with summands isomorphic to Q and Prilfer groups Q /Z for various prime numbers p, and the cardinality of the set of summands of each type is uniquely determined. Moreover, if a divisible group A is a subgroup of an abelian group G then A admits a direct complement: a subgroup C of G such that G = A ® C. Thus divisible groups are injective modules in the category of abelian groups, and conversely, every injective abelian group is divisible (Baer's criterion). An abelian group without non-zero divisible subgroups is called reduced. Two important special classes of infinite abelian groups with diametrically opposite properties are torsion groups and torsion-free groups, examplified by the groups Q/Z (periodic) and Q (torsion-free). Torsion groups An abelian group is called periodic or torsion if every element has finite order. A direct sum of finite cyclic groups is periodic. Although the converse statement is not true in general, some special cases are known. The first and second Priifer theorems state that if A is a periodic group and either it has bounded exponent, i.e. «A = for some natural number n, or if A is countable and the /^-heights of the elements of A are finite for each p, then A is isomorphic to a direct sum of finite cyclic groups. The cardinality of the set of direct summands isomorphic to Z/p m Z in such a decomposition is an invariant of A. These theorems were later subsumed in the Kulikov criterion. In a different direction, Helmut Ulm found an extension of the second Priifer theorem to countable abelian p-groups with elements of infinite height: those groups are completely classified by means of their Ulm invariants. Torsion-free and mixed groups An abelian group is called torsion-free if every non-zero element has infinite order. Several classes of torsion-free abelian groups have been extensively studied: • Free abelian groups, i.e. arbitrary direct sums of Z • Cotorsion and algebraically compact torsion-free groups such as the p-adic integers • Slender groups An abelian group that is neither periodic nor torsion-free is called mixed. If A is an abelian group and T(A) is its torsion subgroup then the factor group AIT{A) is torsion-free. However, in general the torsion subgroup is not a direct summand of A, so the torsion-free factor cannot be realized as a subgroup of A and A is not isomorphic to T{A) ® AIT{A). Thus the theory of mixed groups involves more than simply combining the results about periodic and torsion-free groups. Invariants and classification One of the most basic invariants of an infinite abelian group A is its rank: the cardinality of the maximal linearly independent subset of A. Abelian groups of rank are precisely the periodic groups, while torsion-free abelian groups of rank 1 are necessarily subgroups of Q and can be completely described. More generally, a torsion-free abelian group of finite rank r is a subgroup of Q r . On the other hand, the group of p-adic integers Z is a torsion-free abelian group of infinite Z-rank and the groups Z " with different n are non-isomorphic, so this invariant does not even fully capture properties of some familiar groups. Abelian group 73 The classification theorems for finitely generated, divisible, countable periodic, and rank 1 torsion-free abelian groups explained above were all obtained before 1950 and form a foundation of the classification of more general infinite abelian groups. Important technical tools used in classification of infinite abelian groups are pure and basic subgroups. Introduction of various invariants of torsion-free abelian groups has been one avenue of further progress. See the books by Irving Kaplansky, Laszlo Fuchs, Phillip Griffiths, and David Arnold, as well as the proceedings of the conferences on Abelian Group Theory published in Lecture Notes in Mathematics for more recent results. Additive groups of rings The additive group of a ring is an abelian group, but not all abelian groups are additive groups of rings. Some important topics in this area of study are: • Tensor product • Corner's results on countable torsion-free groups • Shelah's work to remove cardinality restrictions Relation to other mathematical topics Many large abelian groups possess a natural topology, which turns them into topological groups. The collection of all abelian groups, together with the homomorphisms between them, forms the category Ab, the prototype of an abelian category. Nearly all well-known algebraic structures other than Boolean algebras, are undecidable. Hence it is surprising that Tarski's student Szmielew (1955) proved that the first order theory of abelian groups, unlike its nonabelian counterpart, is decidable. This decidability, plus the fundamental theorem of finite abelian groups described above, highlight some of the successes in abelian group theory, but there are still many areas of current research: • Amongst torsion-free abelian groups of finite rank, only the finitely generated case and the rank 1 case are well understood; • There are many unsolved problems in the theory of infinite-rank torsion-free abelian groups; • While countable torsion abelian groups are well understood through simple presentations and Ulm invariants, the case of countable mixed groups is much less mature. • Many mild extensions of the first order theory of abelian groups are known to be undecidable. • Finite abelian groups remain a topic of research in computational group theory. Moreover, abelian groups of infinite order lead, quite surprisingly, to deep questions about the set theory commonly assumed to underlie all of mathematics. Take the Whitehead problem: are all Whitehead groups of infinite order also free abelian groups? In the 1970s, Saharon Shelah proved that the Whitehead problem is: • Undecidable in ZFC, the conventional axiomatic set theory from which nearly all of present day mathematics can be derived. The Whitehead problem is also the first question in ordinary mathematics proved undecidable in ZFC; • Undecidable even if ZFC is augmented by taking the generalized continuum hypothesis as an axiom; • Decidable if ZFC is augmented with the axiom of constructibility (see statements true in L). Abelian group 74 A note on the typography Among mathematical adjectives derived from the proper name of a mathematician, the word "abelian" is rare in that it is often spelled with a lowercase a, rather than an uppercase A, indicating how ubiquitous the concept is in modern mathematics. See also Abelianization Class field theory Commutator subgroup Elementary abelian group Pontryagin duality Pure injective module Pure projective module Notes [1] Jacobson (2009), p. 41 [2] For example, Q/Z = 7 Q /Z . p p p [3] Countability assumption in the second Priifer theorem cannot be removed: the torsion subgroup of the direct product of the cyclic groups Z/p m Z for all natural m is not a direct sum of cyclic groups. [4] Abel Prize Awarded: The Mathematicians' Nobel (http://www.maa.org/devlin/devlin_04_04.html) References Cox, David (2004) Galois Theory. Wiley-Interscience. Hoboken, NJ. xx+559 pp. MR21 19052 Fuchs, Laszlo (1970) Infinite abelian groups, Vol. I. Pure and Applied Mathematics, Vol. 36. New York— London: Academic Press, xi+290 pp. MR0255673 (1973) Infinite abelian groups, Vol. II. Pure and Applied Mathematics. Vol. 36-11. New York— London: Academic Press, ix+363 pp. MR0349869 Griffith, Phillip A. (1970). Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 0-226-30870-7. I.N. Herstein (1975), Topics in Algebra, 2nd edition (John Wiley and Sons, New York) ISBN 0-471-02371-X Hillar, Christopher and Rhea, Darren (2007), Automorphisms of finite abelian groups. Amer. Math. Monthly 114, no. 10, 917-923. arXiv:0605185. Jacobson, Nathan (2009). Basic algebra. 1 (2nd ed.). Dover. ISBN 978-0-486-47189-1.. Szmielew, Wanda (1955) "Elementary properties of abelian groups," Fundamenta Mathematica 41: 203-71. Group algebra 75 Group algebra In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra), such that representations of the algebra are related to representations of the group. As such, they are similar to the group ring associated to a discrete group. Group algebras of topological groups: C (G) For the purposes of functional analysis, and in particular of harmonic analysis, one wishes to carry over the group ring construction to topological groups G. In case G is a locally compact Hausdorff group, G carries an essentially unique left-invariant countably additive Borel measure \i called Haar measure. Using the Haar measure, one can define a convolution operation on the space C (G) of complex-valued functions on G with compact support; C (G) can then be given any of various norms and the completion will be a group algebra. To define the convolution operation, let/ and g be two functions in C (G). For t in G, define [/*<?](*)= f f(s)g(s-H)dfi(s). JG The fact that/* g is continuous is immediate from the dominated convergence theorem. Also Support(/ * g) C Support(/) ■ Support^) C (G) also has a natural involution defined by: r(,) = 7(^)A( s - 1 ) where A is the modular function on G. With this involution, it is a *-algebra. Theorem. If C (G) is given the norm := / |/(s) |d/i(s), it becomes is an involutive normed algebra with an approximate identity. JG The approximate identity can be indexed on a neighborhood basis of the identity consisting of compact sets. Indeed if V is a compact neighborhood of the identity, let/ be a non-negative continuous function supported in V such that L fv{g)dn(g) = 1. v Then {/ } is an approximate identity. A group algebra can only have an identity, as opposed to just approximate identity, if and only if the topology on the group is the discrete topology. Note that for discrete groups, C (G) is the same thing as the complex group ring CG. The importance of the group algebra is that it captures the unitary representation theory of G as shown in the following Theorem. Let G be a locally compact group. If U is a strongly continuous unitary representation of G on a Hilbert space H, then M/)= [ f(g)u(g)dfi( 9 ) JG is a non-degenerate bounded *-representation of the normed algebra C (G). The map C/ 1 — > 7T[/ is a bijection between the set of strongly continuous unitary representations of G and non-degenerate bounded * -representations of C (G). This bijection respects unitary equivalence and strong containment. In particular, jt is irreducible if and only if U is irreducible. Non-degeneracy of a representation n of C (G) on a Hilbert space H means that W)f:/GC (G),eG^} Group algebra 76 is dense in H . 71 The convolution algebra L (G) It is a standard theorem of measure theory that the completion of C (G) in the L (G) norm is isomorphic to the space l c L (G) of equivalence classes of functions which are integrable with respect to the Haar measure, where, as usual, two functions are regarded as equivalent if and only if they differ on a set of Haar measure zero. Theorem. L (G) is a Banach *-algebra with the convolution product and involution defined above and with the L norm. L (G) also has a bounded approximate identity. The group C* -algebra C*(G) Let C[G] be the group ring of a discrete group G. For a locally compact group G, the group C*-algebra C*(G) of G is defined to be the C*-enveloping algebra of L (G), i.e. the completion of C (G) with respect to the largest C*-norm: ll/llc^BUpH/JU, 7T where it ranges over all non-degenerate * -representations of C (G) on Hilbert spaces. When G is discrete, it follows from the triangle inequality that, for any such it, it(f) < l|/ll . So the norm is well-defined. It follows from the definition that C*(G) has the following universal property: any *-homomorphism from C[G] to some B( 7Y ) ( me C*-algebra of bounded operators on some Hilbert space "}-[ ) factors through the inclusion map C[G] ^-s- C* (G). max The reduced group C*-algebra C (G) The reduced group C*-algebra C (G) is the completion of C (G) with respect to the norm ||/||c?; := sup{||/ *g\\ 2 : ||ff|| 2 = 1}, where 2 2 * is the L norm. Since the completion of C (G) with regard to the L norm is a Hilbert space, the C norm is the norm c 2 r of the bounded operator "convolution by/' acting on L (G) and thus a C*-norm. * 2 Equivalently, C (G) is the C*-algebra generated by the image of the left regular representation on I (G). In general, C (G) is a quotient of C (G). The reduced group C*-algebra is isomorphic to the non-reduced group C*-algebra defined above if and only if G is amenable. von Neumann algebras associated to groups The group von Neumann algebra W*(G) of G is the enveloping von Neumann algebra of C*(G). 2 For a discrete group G, we can consider the Hilbert space I (G) for which G is an orthonormal basis. Since G 2 operates on / (G) by permuting the basis vectors, we can identify the complex group ring CG with a subalgebra of 2 the algebra of bounded operators on I (G). The weak closure of this subalgebra, NG, is a von Neumann algebra. The center of NG can be described in terms of those elements of G whose conjugacy class is finite. In particular, if the identity element of G is the only group element with that property (that is, G has the infinite conjugacy class property), the center of NG consists only of complex multiples of the identity. NG is isomorphic to the hyperfinite type II factor if and only if G is countable, amenable, and has the infinite conjugacy class property. Group algebra 77 References • J, Dixmier, C* algebras, ISBN 0-7204-0762-1 • A. A. Kirillov, Elements of the theory of representations, ISBN 0-387-07476-7 • A.I. Shtern (2001), "Group algebra of a locally compact group" , in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104 This article incorporates material from Group $C A *$-algebra on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. • L. H. Loomis, "Abstract Harmonic Analysis", ASIN B0007FUU30 References [1] http://eom.springer.de/G/g045230.htm Cayley's theorem In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group on G. This can be understood as an example of the group action of G on the elements of G. A permutation of a set G is any bijective function taking G onto G; and the set of all such functions forms a group under function composition, called the symmetric group on G, and written as Sym(G). Cayley's theorem puts all groups on the same footing, by considering any group (including infinite groups such as (/?,+)) as a permutation group of some underlying set. Thus, theorems which are true for permutation groups are true for groups in general. History Although Burnside attributes the theorem to Jordan, Eric Nummela nonetheless argues that the standard name — "Cayley's Theorem" — is in fact appropriate. Cayley, in his original 1854 paper, showed that the correspondence in the theorem is one-to-one, but he failed to explicitly show it was a homomorphism (and thus an isomorphism). However, Nummela notes that Cayley made this result known to the mathematical community at the time, thus predating Jordan by 16 years or so. Proof of the theorem Where g is any element of G, consider the function/ : G — > G, defined by/ (x) = g*x. By the existence of inverses, this function has a two-sided inverse, /„-i. So multiplication by g acts as a bijective function. Thus, / is a g permutation of G, and so is a member of Sym(G). The set K = [f : g in G} is a subgroup of Sym(G) which is isomorphic to G. The fastest way to establish this is to consider the function T : G — > Sym(G) with T(g) = f for every g in G. T is a group homomorphism because (using "•" for composition in Sym(G)): (f g 'f h )(x) =f g (f h (x)) =f g (h*x) = g*(h*x) = (g*h)*x =f (g , h) (x), for all x in G, and hence: T ( g ) . m =f g 'f h =f (g . th) =T(g*h). The homomorphism T is also injective since T(g) = id (the identity element of Sym(G)) implies that g*x = x for all x in G, and taking x to be the identity element e of G yields g = g*e = e. Alternatively, T is also injective since, if g*x=g'*x implies g=g' (by post-multiplying with the inverse of x, which exists because G is a group). Cayley's theorem 78 Thus G is isomorphic to the image of T, which is the subgroup K. Tis sometimes called the regular representation of G. Alternative setting of proof An alternative setting uses the language of group actions. We consider the group Q as a G-set, which can be shown to have permutation representation, say (j) . Firstly, suppose G = G/H with H = {e} • Then the group action is 5- e by classification of G-orbits (also known as the orbit-stabilizer theorem). Now, the representation is faithful if (j) is injective, that is, if the kernel of (j) is trivial. Suppose Q € ker (j) Then, g = g.e = (j)(g). e by the equivalence of the permutation representation and the group action. But since Q € ker 4> , (f>(g) = e and thus ker (j) is trivial. Then im (j) < G and thus the result follows by use of the first isomorphism theorem. Remarks on the regular group representation The identity group element corresponds to the identity permutation. All other group elements correspond to a permutation that does not leave any element unchanged. Since this also applies for powers of a group element, lower than the order of that element, each element corresponds to a permutation which consists of cycles which are of the same length: this length is the order of that element. The elements in each cycle form a left coset of the subgroup generated by the element. Examples of the regular group representation Z 2 = {0,1} withE permutation (12). Z 3 = {0,1,2} with permutation (123), and group element 2 to permutation (132). E.g. 1 + 1=2 corresponds to (123)(123)=(132). Z = {0,1,2,3} with addition modulo 4; the elements correspond to e, (1234), (13)(24), (1432). The elements of Klein four-group {e, a, b, c} correspond to e, (12)(34), (13)(24), and (14)(23). S (dihedral groi group elements: Z = {0,1} with addition modulo 2; group element corresponds to the identity permutation e, group element 1 to Z = {0,1,2} with addition modulo 3; group element corresponds to the identity permutation e, group element 1 to S (dihedral group of order 6) is the group of all permutations of 3 objects, but also a permutation group of the 6 * e a b c d f permutation e e a b c d f e a a e d f b c (12)(35)(46) b b f e d c a (13)(26)(45) c c d f e a b (14)(25)(36) d d c a b f e (156)(243) f f b c a e d (165X234) Cayley's theorem 79 See also • Containment order, a similar result in order theory • Frucht's theorem, every group is the automorphism group of a graph • Yoneda lemma, an analogue of Cayley's theorem in category theory Notes [1] Jacobson (2009), p. 38. [2] Jacobson (2009), p. 72, ex. 1. [3] Jacobson (2009), p. 31. [4] Burnside, William (1911), Theory of Groups of Finite Order (2 ed.), Cambridge, ISBN 0486495752 [5] Jordan, Camille (1870), Traite des substitutions et des equations algebriques, Paris: Gaucher- Villars [6] Nummela, Eric (1980), "Cayley's Theorem for Topological Groups" (http://jstor.org/stable/2321608), American Mathematical Monthly (Mathematical Association of America) 87 (3): 202-203, doi: 10.2307/2321608, [7] Cayley, Arthur (1854), "On the theory of groups as depending on the symbolic equation 8"=1", Phil. Mag. 7 (4): 40-47 References • Jacobson, Nathan (2009), Basic algebra (2nd ed.), Dover, ISBN 978-0-486-47189-1. 80 Special Algebras, Operator Algebra and Quantum Algebra Lie algebra In mathematics, a Lie algebra (pronounced I'M'.I ("lee"), not /'lal/ ("lye")) is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Definition and first properties A Lie algebra is a vector space 0over some field F together with a binary operation [■, ■] [t] =fl Xfl-^fl called the Lie bracket, which satisfies the following axioms: • Bilinearity: [ax + by,z] — a[x, z] + b[y, z], [z, ax + by] = a[z, x] + b[z, y] for all scalars a, b in F and all elements x, y, z in 0. • Alternating on : [x,x] — for all x in 0. This implies anticommutativity, or skew-symmetry (in fact the conditions are equivalent for any Lie algebra over any field whose characteristic is not 2): [x,y] = -[y, x ] for all elements x, y in . • The Jacobi identity: [z, [y, z]] + [y, [z, x\] + [z, [x, y]] = for allx, y, zin 0. For any associative algebra A with multiplication * , one can construct a Lie algebra L{A). As a vector space, L{A) is the same as A. The Lie bracket of two elements of L{A) is defined to be their commutator in A: [a,b] — a*b — b * a. The associativity of the multiplication * in A implies the Jacobi identity of the commutator in L{A). In particular, the associative algebra of n x n matrices over a field F gives rise to the general linear Lie algebra g[ (i^Y The associative algebra A is called an enveloping algebra of the Lie algebra L(A). It is known that every Lie algebra can be embedded into one that arises from an associative algebra in this fashion. See universal enveloping algebra. Lie algebra 8 1 Homomorphisms, subalgebras, and ideals The Lie bracket is not an associative operation in general, meaning that \\x y] zlneed not equal \x [y, z]]- Nonetheless, much of the terminology that was developed in the theory of associative rings or associative algebras is commonly applied to Lie algebras. A subspace f) C gthat is closed under the Lie bracket is called a Lie subalgebra. If a subspace / C g satisfies a stronger condition that then / is called an ideal in the Lie algebra g. A Lie algebra in which the commutator is not identically zero and which has no proper ideals is called simple. A homomorphism between two Lie algebras (over the same ground field) is a linear map that is compatible with the commutators: /:fl->fl', f([x y y]) = [f(x)J(y)l for all elements x and y in 0. As in the theory of associative rings, ideals are precisely the kernels of homomorphisms, given a Lie algebra gand an ideal / in it, one constructs the factor algebra q/I , and the first isomorphism theorem holds for Lie algebras. Given two Lie algebras gand g', their direct sum is the vector space g © g' consisting of the pairs (x,x r ), X G g, x' G g', with the operation Examples • Any vector space V endowed with the identically zero Lie bracket becomes a Lie algebra. Such Lie algebras are called abelian, cf. below. Any one-dimensional Lie algebra over a field is abelian, by the antisymmetry of the Lie bracket. 3 • The three-dimensional Euclidean space R with the Lie bracket given by the cross product of vectors becomes a three-dimensional Lie algebra. • The Heisenberg algebra is a three-dimensional Lie algebra with generators (see also the definition at Generating set): / 1 \ / \ / 1 i=0 , y= I 1 , z= I \ o / \o 0/ \ whose commutation relations are [z, y] = z, [x, z] = 0, [y, z] = 0. It is explicitly exhibited as the space of 3x3 strictly upper-triangular matrices. • The subspace of the general linear Lie algebra g[ (]?} consisting of matrices of trace zero is a subalgebra, the special linear Lie algebra, denoted s\ n (F\ • Any Lie group G defines an associated real Lie algebra g = Lie(G)- The definition in general is somewhat technical, but in the case of real matrix groups, it can be formulated via the exponential map, or the matrix exponent. The Lie algebra gconsists of those matrices X for which exp(iX) G G for all real numbers t. The Lie bracket of gis given by the commutator of matrices. As a concrete example, consider the special linear group SL(«,R), consisting of all n x n matrices with real entries and determinant 1 . This is a matrix Lie group, and its Lie algebra consists of all n x n matrices with real entries and trace 0. • The real vector space of all n x n skew-hermitian matrices is closed under the commutator and forms a real Lie algebra denoted u(n)- This is the Lie algebra of the unitary group U(n). • An important class of infinite-dimensional real Lie algebras arises in differential topology. The space of smooth vector fields on a differentiable manifold M forms a Lie algebra, where the Lie bracket is defined to be the Lie algebra 82 commutator of vector fields. One way of expressing the Lie bracket is through the formalism of Lie derivatives, which identifies a vector field X with a first order partial differential operator L acting on smooth functions by letting L if) be the directional derivative of the function/in the direction of X. The Lie bracket [X,Y] of two vector fields is the vector field defined through its action on functions by the formula: L[x,Y]f — L x {L Y f) - L Y (L x f)- This Lie algebra is related to the pseudogroup of diffeomorphisms of M. • The commutation relations between the x, y, and z components of the angular momentum operator in quantum mechanics form a representation of a complex three-dimensional Lie algebra, which is the complexification of the Lie algebra so(3) of the three-dimensional rotation group: [L x ,L y ] = ihL z [L y ,L z ] — ihL x [L Z ,L X ] =ihL y • Kac— Moody algebra is an example of an infinite-dimensional Lie algebra. Structure theory and classification Every finite-dimensional real or complex Lie algebra has a faithful representation by matrices (Ado's theorem). Lie's fundamental theorems describe a relation between Lie groups and Lie algebras. In particular, any Lie group gives rise to a canonically determined Lie algebra (concretely, the tangent space at the identity), and conversely, for any Lie algebra there is a corresponding connected Lie group (Lie's third theorem). This Lie group is not determined uniquely, however, any two connected Lie groups with the same Lie algebra are locally isomorphic, and in particular, have the same universal cover. For instance, the special orthogonal group SO(3) and the special unitary group SU(2) give rise to the same Lie algebra, which is isomorphic to R with the cross-product, and SU(2) is a simply-connected twofold cover of SO(3). Real and complex Lie algebras can be classified to some extent, and this is often an important step toward the classification of Lie groups. Abelian, nilpotent, and solvable Analogously to abelian, nilpotent, and solvable groups, defined in terms of the derived subgroups, one can define abelian, nilpotent, and solvable Lie algebras. A Lie algebra gis abelian if the Lie bracket vanishes, i.e. [x,y] = 0, for all x and y in Q. Abelian Lie algebras correspond to commutative (or abelian) connected Lie groups such as vector spaces J( n OT tori T"",and are all of the form t n , meaning an n-dimensional vector space with the trivial Lie bracket. A more general class of Lie algebras is defined by the vanishing of all commutators of given length. A Lie algebra is nilpotent if the lower central series > [0,0] > [[fl,0],fl] > [[[0,0], 0L0] > ■•■ becomes zero eventually. By Engel's theorem, a Lie algebra is nilpotent if and only if for every u in flthe adjoint endomorphism ad(u) : — >■ 3, ad(u)v = [u, v] is nilpotent. More generally still, a Lie algebra 0is said to be solvable if the derived series: > [0,0] > [[0,0], [0,0]] > [[[0,0], [0,0]], [[0,0], [0,0]]] > ■■■ becomes zero eventually. Every finite-dimensional Lie algebra has a unique maximal solvable ideal, called its radical. Under the Lie correspondence, nilpotent (respectively, solvable) connected Lie groups correspond to nilpotent (respectively, Lie algebra 83 solvable) Lie algebras. Simple and semisimple A Lie algebra is "simple" if it has no non-trivial ideals and is not abelian. A Lie algebra gis called semisimple if its radical is zero. Equivalently, gis semisimple if it does not contain any non-zero abelian ideals. In particular, a simple Lie algebra is semisimple. Conversely, it can be proven that any semisimple Lie algebra is the direct sum of its minimal ideals, which are canonically determined simple Lie algebras. The concept of semisimplicity for Lie algebras is closely related with the complete reducibility of their representations. When the ground field F has characteristic zero, semisimplicity of a Lie algebra gover F is equivalent to the complete reducibility of all finite-dimensional representations of g.An early proof of this statement proceeded via connection with compact groups (Weyl's unitary trick), but later entirely algebraic proofs were found. Classification In many ways, the classes of semisimple and solvable Lie algebras are at the opposite ends of the full spectrum of the Lie algebras. The Levi decomposition expresses an arbitrary Lie algebra as a semidirect sum of its solvable radical and a semisimple Lie algebra, almost in a canonical way. Semisimple Lie algebras over an algebraically closed field have been completely classified through their root systems. The classification of solvable Lie algebras is a 'wild' problem, and cannot be accomplished in general. Cartan's criterion gives conditions for a Lie algebra to be nilpotent, solvable, or semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on ^defined by the formula K(u, v) — tr(ad(u)ad(i))), where tr denotes the trace of a linear operator. A Lie algebra gis semisimple if and only if the Killing form is nondegenerate. A Lie algebra gis solvable if and only if K(g, [g, gl) = 0. Relation to Lie groups Although Lie algebras are often studied in their own right, historically they arose as a means to study Lie groups. Given a Lie group, a Lie algebra can be associated to it either by endowing the tangent space to the identity with the differential of the adjoint map, or by considering the left-invariant vector fields as mentioned in the examples. This association is functorial, meaning that homomorphisms of Lie groups lift to homomorphisms of Lie algebras, and various properties are satisfied by this lifting: it commutes with composition, it maps Lie subgroups, kernels, quotients and cokernels of Lie groups to subalgebras, kernels, quotients and cokernels of Lie algebras, respectively. The functor which takes each Lie group to its Lie algebra and each homomorphism to its differential is a faithful and exact functor. This functor is not invertible; different Lie groups may have the same Lie algebra, for example SO(3) and SU(2) have isomorphic Lie algebras. Even worse, some Lie algebras need not have any associated Lie group. Nevertheless, when the Lie algebra is finite-dimensional, there is always at least one Lie group whose Lie algebra is the one under discussion, and a preferred Lie group can be chosen. Any finite-dimensional connected Lie group has a universal cover. This group can be constructed as the image of the Lie algebra under the exponential map. More generally, we have that the Lie algebra is homeomorphic to a neighborhood of the identity. But globally, if the Lie group is compact, the exponential will not be injective, and if the Lie group is not connected, simply connected or compact, the exponential map need not be surjective. If the Lie algebra is infinite-dimensional, the issue is more subtle. In many instances, the exponential map is not even locally a homeomorphism (for example, in Diff(S ), one may find diffeomorphisms arbitrarily close to the identity which are not in the image of exp). Furthermore, some infinite-dimensional Lie algebras are not the Lie algebra of any group. Lie algebra 84 The correspondence between Lie algebras and Lie groups is used in several ways, including in the classification of Lie groups and the related matter of the representation theory of Lie groups. Every representation of a Lie algebra lifts uniquely to a representation of the corresponding connected, simply connected Lie group, and conversely every representation of any Lie group induces a representation of the group's Lie algebra; the representations are in one to one correspondence. Therefore, knowing the representations of a Lie algebra settles the question of representations of the group. As for classification, it can be shown that any connected Lie group with a given Lie algebra is isomorphic to the universal cover mod a discrete central subgroup. So classifying Lie groups becomes simply a matter of counting the discrete subgroups of the center, once the classification of Lie algebras is known (solved by Cartan et al. in the semisimple case). Category theoretic definition Using the language of category theory, a Lie algebra can be defined as an object A in Vec, the category of vector spaces together with a morphism [.,.]: A ® A — > A, where <S> refers to the monoidal product of Vec, such that • [-,-]o(id + r AA )=0 • [-,-] O ([-,-] ® id) O (id + <7 + <7 2 )=0 where x (a <S> b) := b <E> a and o is the cyclic permutation braiding (id <E> x ) ° (x ® id). In diagrammatic form: Notes [1] Due to the anticommutativity of the commutator, the notions of a left and right ideal in a Lie algebra coincide. [2] Humphreys p. 2 References • Hall, Brian C. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, 2003. ISBN 0-387-40122-9 • Erdmann, Karin & Wildon, Mark. Introduction to Lie Algebras, 1st edition, Springer, 2006. ISBN 1-84628-040-0 • Humphreys, James E. Introduction to Lie Algebras and Representation Theory, Second printing, revised. Graduate Texts in Mathematics, 9. Springer- Verlag, New York, 1978. ISBN 0-387-90053-5 • Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4 • Kac, Victor G. et al. Course notes for MIT 18. 745: Introduction to Lie Algebras, http://www-math.mit.edu/ ~lesha/7451ec/ • O'Connor, J. J. & Robertson, E.F. Biography of Sophus Lie, MacTutor History of Mathematics Archive, http:// www-history. mcs.st-and.ac.uk/Biographies/Lie.html • O'Connor, J. J. & Robertson, E.F. Biography of Wilhelm Killing, MacTutor History of Mathematics Archive, http://www-history.mcs.st-and.ac.uk/Biographies/Killing.html • Steeb, W.-H. Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra, second edition, World Scientific, 2007, ISBN 978-981-270-809-0 Lie algebra 85 Varadarajan, V. S. Lie Groups, Lie Algebras, and Their Representations, 1st edition. Springer, 2004. ISBN 0-387-90969-9 Lie group In mathematics, a Lie group (pronounced /'li:/: similar to "Lee") is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups. Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations (Differential Galois theory), in much the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations. Overview Lie groups are smooth manifolds and, therefore, can be studied using differential calculus, in contrast with the case of more general topological groups. One of the key ideas in the theory of Lie groups, from Sophus Lie, is to replace the global object, the group, with its local or linearized version, which Lie himself called its "infinitesimal group" and which has since become known as its Lie algebra. Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) 3 of distance-preserving transformations of the Euclidean space R , conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure, where G is a Lie group of "local" symmetries of a manifold. On a "global" level, whenever a Lie group acts on a geometric object, such as a Riemannian or a symplectic manifold, this action provides a measure of rigidity and yields a rich algebraic structure. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry and facilitates analysis on the manifold. Linear actions of Lie groups are especially important, and are studied in representation theory. In the 1940s— 1950s, Ellis Kolchin, Armand Borel and Claude Chevalley realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of algebraic groups defined over an arbitrary field. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, as well as in algebraic geometry. The theory of automorphic forms, an important branch of modern number theory, deals extensively with analogues of Lie groups over adele rings; p-adic Lie groups play an important role, via their connections with Galois representations in number theory. The circle of center and radius 1 in the complex plane is a Lie group with complex multiplication. Lie group 86 Definitions and examples A real Lie group is a group which is also a finite-dimensional real smooth manifold, and in which the group operations of multiplication and inversion are smooth maps. Smoothness of the group multiplication /i:GxG->G /i(x, y) — xy means that \i is a smooth mapping of the product manifold GxG into G. These two requirements can be combined to the single requirement that the mapping (x,y) ^x~ x y be a smooth mapping of the product manifold into G. First examples • The 2x2 real invertible matrices form a group under multiplication, denoted by GL (R): GL 2 (M) = I A = ( a b _, J : det A = ad - be ^ This is a four-dimensional noncompact real Lie group. This group is disconnected; it has two connected components corresponding to the positive and negative values of the determinant. • The rotation matrices form a subgroup of GL (R), denoted by SO (R). It is a Lie group in its own right: specifically, a one-dimensional compact connected Lie group which is diffeomorphic to the circle. Using the rotation angle <fi as a parameter, this group can be parametrized as follows: S0 2 (R) = l( COSlfi - Sm ^L £ l/ 2 4. [ ysm ip cos ip J J Addition of the angles corresponds to multiplication of the elements of SO (R), and taking the opposite angle corresponds to inversion. Thus both multiplication and inversion are differentiable maps. • The orthogonal group also forms an interesting example of a Lie group. All of the previous examples of Lie groups fall within the class of classical groups Related concepts A complex Lie group is defined in the same way using complex manifolds rather than real ones (example: SL -(C)), and similarly one can define a p-adic Lie group over the p-adic numbers. Hilbert's fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples. The answer to this question turned out to be negative: in 1952, Gleason, Montgomery and Zippin showed that if G is a topological manifold with continuous group operations, then there exists exactly one analytic structure on G which turns it into a Lie group (see also Hilbert— Smith conjecture). If the underlying manifold is allowed to be infinite dimensional (for example, a Hilbert manifold) then one arrives at the notion of an infinite-dimensional Lie group. It is possible to define analogues of many Lie groups over finite fields, and these give most of the examples of finite simple groups. The language of category theory provides a concise definition for Lie groups: a Lie group is a group object in the category of smooth manifolds. This is important, because it allows generalization of the notion of a Lie group to Lie supergroups. Lie group 87 More examples of Lie groups Lie groups occur in abundance throughout mathematics and physics. Matrix groups or algebraic groups are (roughly) groups of matrices (for example, orthogonal and symplectic groups), and these give most of the more common examples of Lie groups. Examples • Euclidean space R" with ordinary vector addition as the group operation becomes an n-dimensional noncompact abelian Lie group. • The circle group S consisting of angles mod 2m under addition or, alternately, the complex numbers with absolute value 1 under multiplication. This is a one-dimensional compact connected abelian Lie group. 2 • The group GL (R) of invertible matrices (under matrix multiplication) is a Lie group of dimension n , called the general linear group. It has a closed connected subgroup SL (R), the special linear group, consisting of matrices of determinant 1 which is also a Lie group. • The orthogonal group O (R), consisting of all n x n orthogonal matrices with real entries is an n{n - l)/2-dimensional Lie group. This group is disconnected, but it has a connected subgroup SO (R) of the same dimension consisting of orthogonal matrices of determinant 1, called the special orthogonal group (for n = 3, the rotation group). • The Euclidean group E (R) is the Lie group of all Euclidean motions, i.e., isometric affine maps, of n-dimensional Euclidean space R". • The unitary group U(«) consisting ofnxn unitary matrices (with complex entries) is a compact connected Lie 2 2 group of dimension n . Unitary matrices of determinant 1 form a closed connected subgroup of dimension n - 1 denoted SU(«), the special unitary group. • Spin groups are double covers of the special orthogonal groups, used for studying fermions in quantum field theory (among other things). • The symplectic group Sp (R) consists of all 2« x 2« matrices preserving a nondegenerate skew-symmetric 2fi 2 bilinear form on R (the symplectic form). It is a connected Lie group of dimension 2« + n. The fundamental group of the symplectic group is Z and this fact is related to the theory of Maslov index. • The 3-sphere S forms a Lie group by identification with the set of quaternions of unit norm, called versors. The only other spheres that admit the structure of a Lie group are the 0-sphere S (real numbers with absolute value 1) 1 n and the circle S (complex numbers with absolute value 1). For example, for even n > 1, S is not a Lie group because it does not admit a nonvanishing vector field and so a fortiori cannot be parallelizable as a differentiable manifold. Of the spheres only !>,S,S, and S are parallelizable. The latter carries the structure of a Lie quasigroup (a nonassociative group), which can be identified with the set of unit octonions. • The group of upper triangular n by n matrices is a solvable Lie group of dimension n(n + l)/2. • The Lorentz group and the Poincare group are the groups of linear and affine isometries of the Minkowski space (interpreted as the spacetime of the special relativity). They are Lie groups of dimensions 6 and 10. • The Heisenberg group is a connected nilpotent Lie group of dimension 3, playing a key role in quantum mechanics. • The group U(l)xSU(2)xSU(3) is a Lie group of dimension 1+3+8=12 that is the gauge group of the Standard Model in particle physics. The dimensions of the factors correspond to the 1 photon + 3 vector bosons + 8 gluons of the standard model. • The (3-dimensional) metaplectic group is a double cover of SL (R) playing an important role in the theory of modular forms. It is a connected Lie group that cannot be faithfully represented by matrices of finite size, i.e., a nonlinear group. • The exceptional Lie groups of types G , F , E , E , E have dimensions 14, 52, 78, 133, and 248. There is also a group E of dimension 190. Lie group Constructions There are several standard ways to form new Lie groups from old ones: • The product of two Lie groups is a Lie group. • Any topologically closed subgroup of a Lie group is a Lie group. This is known as Cartan's theorem. • The quotient of a Lie group by a closed normal subgroup is a Lie group. • The universal cover of a connected Lie group is a Lie group. For example, the group R is the universal cover of the circle group S . In fact any covering of a differentiable manifold is also a differentiable manifold, but by specifying universal cover, one guarantees a group structure (compatible with its other structures). Related notions Some examples of groups that are not Lie groups (except in the trivial sense that any group can be viewed as a O-dimensional Lie group, with the discrete topology), are: • Infinite dimensional groups, such as the additive group of an infinite dimensional real vector space. These are not Lie groups as they are not finite dimensional manifolds • Some totally disconnected groups, such as the Galois group of an infinite extension of fields, or the additive group of the p-adic numbers. These are not Lie groups because their underlying spaces are not real manifolds. (Some of these groups are "p-adic Lie groups"). In general, only topological groups having similar local properties to R for some positive integer n can be Lie groups (of course they must also have a differentiable structure) Early history According to the most authoritative source on the early history of Lie groups (Hawkins, p. 1), Sophus Lie himself considered the winter of 1873—1874 as the birth date of his theory of continuous groups. Hawkins, however, suggests that it was "Lie's prodigious research activity during the four-year period from the fall of 1869 to the fall of 1873" that led to the theory's creation {ibid). Some of Lie's early ideas were developed in close collaboration with Felix Klein. Lie met with Klein every day from October 1869 through 1872: in Berlin from the end of October 1869 to the end of February 1870, and in Paris, Gottingen and Erlangen in the subsequent two years {ibid, p. 2). Lie stated that all of the principal results were obtained by 1884. But during the 1870s all his papers (except the very first note) were published in Norwegian journals, which impeded recognition of the work throughout the rest of Europe {ibid, p. 76). In 1884 a young German mathematician, Friedrich Engel, came to work with Lie on a systematic treatise to expose his theory of continuous groups. From this effort resulted the three-volume Theorie der Transformations gruppen, published in 1888, 1890, and 1893. Lie's ideas did not stand in isolation from the rest of mathematics. In fact, his interest in the geometry of differential equations was first motivated by the work of Carl Gustav Jacobi, on the theory of partial differential equations of first order and on the equations of classical mechanics. Much of Jacobi's work was published posthumously in the 1860s, generating enormous interest in France and Germany (Hawkins, p. 43). Lie's idee fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Evariste Galois had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. Additional impetus to consider continuous groups came from ideas of Bernhard Riemann, on the foundations of geometry, and their further development in the hands of Klein. Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as exemplified by Galois through the algebraic notion of a group; geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi; and the new understanding of geometry that emerged in the works of Pliicker, Mobius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject. Lie group Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups, a major stride in the development of their structure theory, which was to have a profound influence on subsequent development of mathematics, was made by Wilhelm Killing, who in 1888 published the first paper in a series entitled Die Zusammensetzung der stetigen endlichen Transformationsgruppen {The composition of continuous finite transformation groups) (Hawkins, p. 100). The work of Killing, later refined and generalized by Elie Cartan, led to classification of semisimple Lie algebras, Cartan's theory of symmetric spaces, and Hermann Weyl's description of representations of compact and semisimple Lie groups using highest weights. Weyl brought the early period of the development of the theory of Lie groups to fruition, for not only did he classify irreducible representations of semisimple Lie groups and connect the theory of groups with quantum mechanics, but he also put Lie's theory itself on firmer footing by clearly enunciating the distinction between Lie's infinitesimal groups (i.e., Lie algebras) and the Lie groups proper, and began investigations of topology of Lie groups (Borel (2001), ). The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by Claude Chevalley. The concept of a Lie group, and possibilities of classification Lie groups may be thought of as smoothly varying families of symmetries. Examples of symmetries include rotation about an axis. What must be understood is the nature of 'small' transformations, e.g., rotations through tiny angles, that link nearby transformations. The mathematical object capturing this structure is called a Lie algebra (Lie himself called them "infinitesimal groups"). It can be defined because Lie groups are manifolds, so have tangent spaces at each point. The Lie algebra of any compact Lie group (very roughly: one for which the symmetries form a bounded set) can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones. The structure of an abelian Lie algebra is mathematically uninteresting (since the Lie bracket is identically zero); the interest is in the simple summands. Hence the question arises: what are the simple Lie algebras of compact groups? It turns out that they mostly fall into four infinite families, the "classical Lie algebras" A , B , C and D , which have simple descriptions n n n n in terms of symmetries of Euclidean space. But there are also just five "exceptional Lie algebras" that do not fall into any of these families. E is the largest of these. 8 Properties • The diffeomorphism group of a Lie group acts transitively on the Lie group • Every Lie group is parallelizable, and hence an orientable manifold (there is a bundle isomorphism between its tangent bundle and the product of itself with the tangent space at the identity) Types of Lie groups and structure theory Lie groups are classified according to their algebraic properties (simple, semisimple, solvable, nilpotent, abelian), their connectedness (connected or simply connected) and their compactness. • Compact Lie groups are all known: they are finite central quotients of a product of copies of the circle group S and simple compact Lie groups (which correspond to connected Dynkin diagrams). • Any simply connected solvable Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional. Solvable groups are too messy to classify except in a few small dimensions. • Any simply connected nilpotent Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with l's on the diagonal of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional. Like solvable groups, nilpotent groups are too messy to classify except in a few small dimensions. Lie group 90 • Simple Lie groups are sometimes defined to be those that are simple as abstract groups, and sometimes defined to be connected Lie groups with a simple Lie algebra. For example, SL (R) is simple according to the second definition but not according to the first. They have all been classified (for either definition). • Semisimple Lie groups are Lie groups whose Lie algebra is a product of simple Lie algebras. They are central extensions of products of simple Lie groups. The identity component of any Lie group is an open normal subgroup, and the quotient group is a discrete group. The universal cover of any connected Lie group is a simply connected Lie group, and conversely any connected Lie group is a quotient of a simply connected Lie group by a discrete normal subgroup of the center. Any Lie group G can be decomposed into discrete, simple, and abelian groups in a canonical way as follows. Write G for the connected component of the identity con L J G for the largest connected normal solvable subgroup G .. for the largest connected normal nilpotent subgroup so that we have a sequence of normal subgroups ICG CG CG C G. nil sol con Then GIG is discrete con G IG , is a central extension of a product of simple connected Lie groups. con sol c L •=> l G IG is abelian. A connected abelian Lie group is isomorphic to a product of copies of R and the circle group S . G /l is nilpotent, and therefore its ascending central series has all quotients abelian. This can be used to reduce some problems about Lie groups (such as finding their unitary representations) to the same problems for connected simple groups and nilpotent and solvable subgroups of smaller dimension. The Lie algebra associated with a Lie group To every Lie group, we can associate a Lie algebra, whose underlying vector space is the tangent space of G at the identity element, which completely captures the local structure of the group. Informally we can think of elements of the Lie algebra as elements of the group that are "infinitesimally close" to the identity, and the Lie bracket is something to do with the commutator of two such infinitesimal elements. Before giving the abstract definition we give a few examples: • The Lie algebra of the vector space R" is just R n with the Lie bracket given by [A, B] = 0. (In general the Lie bracket of a connected Lie group is always if and only if the Lie group is abelian.) • The Lie algebra of the general linear group GL (R) of invertible matrices is the vector space M (R) of square matrices with the Lie bracket given by [A,B]=AB-BA. If G is a closed subgroup of GL (R) then the Lie algebra of G can be thought of informally as the matrices m of " 2 M (R) such that 1 + em is in G, where e is an infinitesimal positive number with e = (of course, no such real T number e exists). For example, the orthogonal group O (R) consists of matrices A with AA = 1, so the Lie algebra T T 2 consists of the matrices m with (1 + Em)(l + em) = 1, which is equivalent to m + m =0 because e = 0. • Formally, when working over the reals, as here, this is accomplished by considering the limit as e — > 0; but the "infinitesimal" language generalizes directly to Lie groups over general rings. The concrete definition given above is easy to work with, but has some minor problems: to use it we first need to represent a Lie group as a group of matrices, but not all Lie groups can be represented in this way, and it is not Lie group 9 1 obvious that the Lie algebra is independent of the representation we use. To get round these problems we give the general definition of the Lie algebra of any Lie group (in 4 steps): 1 . Vector fields on any smooth manifold M can be thought of as derivations X of the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracket [X, Y] = XY - YX, because the Lie bracket of any two derivations is a derivation. 2. If G is any group acting smoothly on the manifold M, then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra. 3. We apply this construction to the case when the manifold M is the underlying space of a Lie group G, with G acting on G = M by left translations L (h) = gh. This shows that the space of left invariant vector fields (vector fields satisfying L t X =X for every h in G, where L ^ denotes the differential of L ) on a Lie group is a Lie 8 rt gll g g algebra under the Lie bracket of vector fields. 4. Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold. Specifically, the left invariant extension of an element v of the tangent space at the identity is the vector field defined by v A = L t v. This identifies the tangent space T at the 8 8 & identity with the space of left invariant vector fields, and therefore makes the tangent space at the identity into a Lie algebra, called the Lie algebra of G, usually denoted by a Fraktur 0. Thus the Lie bracket on 0is given explicitly by [v, w] = [v A , vv A ] . This Lie algebra 0is finite-dimensional and it has the same dimension as the manifold G. The Lie algebra of G determines G up to "local isomorphism", where two Lie groups are called locally isomorphic if they look the same near the identity element. Problems about Lie groups are often solved by first solving the corresponding problem for the Lie algebras, and the result for groups then usually follows easily. For example, simple Lie groups are usually classified by first classifying the corresponding Lie algebras. We could also define a Lie algebra structure on T using right invariant vector fields instead of left invariant vector fields. This leads to the same Lie algebra, because the inverse map on G can be used to identify left invariant vector fields with right invariant vector fields, and acts as -1 on the tangent space T . The Lie algebra structure on T can also be described as follows: the commutator operation / ^ , -1-1 (x, y)-*xyx y on G x G sends (e, e) to e, so its derivative yields a bilinear operation on T G. This bilinear operation is actually the zero map, but the second derivative, under the proper identification of tangent spaces, yields an operation that satisfies the axioms of a Lie bracket, and it is equal to twice the one defined through left-invariant vector fields. Homomorphisms and isomorphisms If G and H are Lie groups, then a Lie-group homomorphism / : G — > H is a smooth group homomorphism. (It is equivalent to require only that /be continuous rather than smooth.) The composition of two such homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms a category. Two Lie groups are called isomorphic if there exists a bijective homomorphism between them whose inverse is also a homomorphism. Isomorphic Lie groups are essentially the same; they only differ in the notation for their elements. Every homomorphism/ : G — > H of Lie groups induces a homomorphism between the corresponding Lie algebras Q and fj. The association G i— ► fjis a functor (mapping between categories satisfying certain axioms). One version of Ado's theorem is that every finite dimensional Lie algebra is isomorphic to a matrix Lie algebra. For every finite dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra. So every abstract Lie algebra is the Lie algebra of some (linear) Lie group. The global structure of a Lie group is not determined by its Lie algebra; for example, if Z is any discrete subgroup of the center of G then G and GIZ have the same Lie algebra (see the table of Lie groups for examples). A connected Lie group is simple, semisimple, solvable, nilpotent, or abelian if and only if its Lie algebra has the corresponding Lie group 92 property. If we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: for every finite dimensional Lie algebra flover F there is a simply connected Lie group G with gas Lie algebra, unique up to isomorphism. Moreover every homomorphism between Lie algebras lifts to a unique homomorphism between the corresponding simply connected Lie groups. The exponential map The exponential map from the Lie algebra M (R) of the general linear group GL (R) to GL (R) is defined by the usual power series: A 2 A 3 e W (A) = l + A + — + — + ■■■ for matrices A. If G is any subgroup of GL (R), then the exponential map takes the Lie algebra of G into G, so we have an exponential map for all matrix groups. The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group. We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows. Every vector v in ^determines a linear map from R to gtaking 1 to v, which can be thought of as a Lie algebra homomorphism. Because R is the Lie algebra of the simply connected Lie group R, this induces a Lie group homomorphism c : R — > G so that c(s + t)= c(s)c(t) for all s and t. The operation on the right hand side is the group multiplication in G. The formal similarity of this formula with the one valid for the exponential function justifies the definition exp(f) = c(l). This is called the exponential map, and it maps the Lie algebra ginto the Lie group G. It provides a diffeomorphism between a neighborhood of in gand a neighborhood of e in G. This exponential map is a generalization of the exponential function for real numbers (because R is the Lie algebra of the Lie group of positive real numbers with multiplication), for complex numbers (because C is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for matrices (because M (R) with the regular commutator is the Lie algebra of the Lie group GL (R) of all invertible matrices). n Because the exponential map is surjective on some neighbourhood N of e, it is common to call elements of the Lie algebra infinitesimal generators of the group G. The subgroup of G generated by N is the identity component of G. The exponential map and the Lie algebra determine the local group structure of every connected Lie group, because of the Baker— Campbell— Hausdorff formula: there exists a neighborhood U of the zero element of g, such that for u, v in U we have exp(M) exp(v) = exp(M + v + 1/2 [u, v] + 1/12 [[u, v], v] - 1/12 [[u, v], u] - ...) where the omitted terms are known and involve Lie brackets of four or more elements. In case u and v commute, this formula reduces to the familiar exponential law exp(u) exp(v) = exp(M + v). The exponential map from the Lie algebra to the Lie group is not always onto, even if the group is connected (though it does map onto the Lie group for connected groups that are either compact or nilpotent). For example, the exponential map of SL (R) is not surjective. Lie group 93 Infinite dimensional Lie groups Lie groups are often defined to be finite dimensional, but there are many groups that resemble Lie groups, except for being infinite dimensional. The simplest was to define infinite dimensional Lie groups is to model them on Banach spaces, and in this case much of the basic theory is similar to that of finite dimensional lie groups. However this is inadequate for many applications, because many natural examples of infinite dimensional Lie groups are not Banach manifolds. Instead one needs to define Lie groups modeled on more general locally convex topological vector spaces. In this case the relation between the Lie algebra and the Lie group becomes rather subtle, and several results about finite dimensional Lie groups no longer hold. Some of the examples that have been studied include: • The group of diffeomorphisms of a manifold. Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra is (more or less) the Witt algebra, which has a central extension called the Virasoro algebra, used in string theory and conformal field theory. Very little is known about the diffeomorphism groups of manifolds of larger dimension. The diffeomorphism group of spacetime sometimes appears in attempts to quantize gravity. • The group of smooth maps from a manifold to a finite dimensional Lie group is an example of a gauge group (with operation of pointwise multiplication), and is used in quantum field theory and Donaldson theory. If the manifold is a circle these are called loop groups, and have central extensions whose Lie algebras are (more or less) Kac— Moody algebras. • There are infinite dimensional analogues of general linear groups, orthogonal groups, and so on. One important aspect is that these may have simpler topological properties: see for example Kuiper's theorem. Notes [1] Sigurdur Helgason, "Differential Geometry, Lie Groups, and Symmetric Spaces", Academic Press, 1978, page 131. References • Adams, John Frank (1969), Lectures on Lie Groups, Chicago Lectures in Mathematics, Chicago: Univ. of Chicago Press, ISBN 0-226-00527-5. • Borel, Armand (2001), Essays in the history of Lie groups and algebraic groups (http://books.google.com/ books?isbn=082 1802887), History of Mathematics, 21, Providence, R.I.: American Mathematical Society, MR1847105, ISBN 978-0-8218-0288-5 • Bourbaki, Nicolas, Elements of mathematics: Lie groups and Lie algebras. Chapters 1—3 ISBN 3-540-64242-0, Chapters 4-6 ISBN 3-540-42650-7, Chapters 7-9 ISBN 3-540-43405-4 • Chevalley, Claude (1946), Theory of Lie groups, Princeton: Princeton University Press, ISBN 0-691-04990-4. • Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics, 129, New York: Springer- Verlag, MR1 153249, ISBN 978-0-387-97527-6, ISBN 978-0-387-97495-8 • Hall, Brian C. (2003), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, ISBN 0-387-40122-9. • Hawkins, Thomas (2000), Emergence of the theory of Lie groups (http://books.google.com/ books?isbn=978-0-387-98963-l), Sources and Studies in the History of Mathematics and Physical Sciences, Berlin, New York: Springer- Verlag, MR1771134, ISBN 978-0-387-98963-1 Borel's review (http://www.jstor. org/stable/2695575) • Knapp, Anthony W. (2002), Lie Groups Beyond an Introduction, Progress in Mathematics, 140 (2nd ed.), Boston: Birkhauser, ISBN 0-8176-4259-5. • Rossmann, Wulf (2001), Lie Groups: An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford University Press, ISBN 978-0198596837. The 2003 reprint corrects several typographical Lie group 94 mistakes. • Serre, Jean-Pierre (1965), Lie Algebras and Lie Groups: 1964 Lectures given at Harvard University, Lecture notes in mathematics, 1500, Springer, ISBN 3-540-55008-9. • Steeb, Willi-Hans (2007), Continuous Symmetries, Lie algebras, Differential Equations and Computer Algebra: second edition, World Scientific Publishing, ISBN 981-270-809-X. Affine Lie algebra In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. It is a Kac— Moody algebra whose generalized Cartan matrix is positive semi-definite and has corank 1. From purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite dimensional, semisimple Lie algebras is much better understood than that of general Kac— Moody algebras. As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities. Affine Lie algebras play an important role in string theory and conformal field theory due to the way they are constructed: starting from a simple Lie algebra 0, one considers the loop algebra, Lg, formed by the -valued functions on a circle (interpreted as the closed string) with pointwise commutator. The affine Lie algebra Sis obtained by adding one extra dimension to the loop algebra and modifying a commutator in a non-trivial way, which physicists call a quantum anomaly. The point of view of string theory helps to understand many deep properties of affine Lie algebras, such as the fact that the characters of their representations are given by modular forms. Affine Lie algebras from simple Lie algebras Construction If 0is a finite dimensional simple Lie algebra, the corresponding affine Lie algebra nis constructed as a central extension of the infinite-dimensional Lie algebra g <g) C[t : t — 1 ]> with one-dimensional center Cc.As a vector space, = ® C[M _1 ] © Cc, where C|£ i ^ "I is the complex vector space of Laurent polynomials in the indeterminate t. The Lie bracket is defined by the formula [a ® t n + ac, b®t m + (3c} = [a, 6] <g) t n+m + {a\b)n8 m+nfi c for all a, b £ 0, a y f3 £E C and ra, m £ Z , where [a, b] is the Lie bracket in the Lie algebra 0and /-|.\ is the Cartan-Killing form on 0. The affine Lie algebra corresponding to a finite-dimensional semisimple Lie algebra is the direct sum of the affine Lie algebras corresponding to its simple summands. Constructing the Dynkin diagrams The Dynkin diagram of each affine Lie algebra consists of that of the corresponding simple Lie algebra plus an additional node, which corresponds to the addition of an imaginary root. Of course, such a node cannot be attached to the Dynkin diagram in just any location, but for each simple Lie algebra there exists a number of possible attachments equal to the cardinality of the group of outer automorphisms of the Lie algebra. In particular, this group always contains the identity element, and the corresponding affine Lie algebra is called an untwisted affine Lie algebra. When the simple algebra admits automorphisms that are not inner automorphisms, one may obtain other Affine Lie algebra 95 Dynkin diagrams and these correspond to twisted affine Lie algebras. Classifying the central extensions The attachment of an extra node to the Dynkin diagram of the corresponding simple Lie algebra corresponds to the following construction. An affine Lie algebra can always be constructed as a central extension of the loop algebra of the corresponding simple Lie algebra. If one wishes to begin instead with a semisimple Lie algebra, then one needs to centrally extend by a number of elements equal to the number of simple components of the semisimple algebra. In physics, one often considers instead the direct sum of a semisimple algebra and an abelian algebra (£ n . In this case one also needs to add n further central elements for the n abelian generators. The second integral cohomology of the loop group of the corresponding simple compact Lie group is isomorphic to the integers. Central extensions of the affine Lie group by a single generator are topologically circle bundles over this free loop group, which are classified by a two-class known as the first Chern class of the fibration. Therefore the central extensions of an affine Lie group are classified by a single parameter k which is called the central charge in the physics literature, where it first appeared. Unitary highest weight representations of the affine compact groups only exist when k is a natural number. More generally, if one considers a semi-simple algebra, there is a central charge for each simple component. Applications They appear naturally in theoretical physics (for example, in conformal field theories such as the WZW model and coset models and even on the worldsheet of the heterotic string), geometry, and elsewhere in mathematics. References • Di Francesco, P.; Mathieu, P.; Senechal, D. (1997), Conformal Field Theory, Springer-Verlag, ISBN 0-387-94785-X • Fuchs, Jurgen (1992), Affine Lie Algebras and Quantum Groups, Cambridge University Press, ISBN 0-521 -484 12-X • Goddard, Peter; Olive, David (1988), Kac-Moody and Virasoro algebras: A Reprint Volume for Physicists, Advanced Series in Mathematical Physics, 3, World Scientific, ISBN 9971-50-419-7 • Kac, Victor (1990), Infinite dimensional Lie algebras (3 ed.), Cambridge University Press, ISBN 0-521-46693-8 • Kohno, Toshitake (1998), Conformal Field Theory and Topology, American Mathematical Society, ISBN 0-8218-2130-X • Pressley, Andrew; Segal, Graeme (1986), Loop groups, Oxford University Press, ISBN 0-19-853535-X Kac— Moody algebra Kac-Moody algebra In mathematics, a Kac— Moody algebra (named for Victor Kac and Robert Moody, who independently discovered them) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix. These algebras form a generalization of finite-dimensional semisimple Lie algebras, and many properties related to the structure of a Lie algebra such as its root system, irreducible representations, and connection to flag manifolds have natural analogues in the Kac— Moody setting. A class of Kac— Moody algebras called affine Lie algebras is of particular importance in mathematics and theoretical physics, especially conformal field theory and the theory of exactly solvable models. Kac discovered an elegant proof of certain combinatorial identities, Macdonald identities, which is based on the representation theory of affine Kac— Moody algebras. H. Garland and Jim Lepowsky demonstrated that Rogers-Ramanujan identities can be derived in a similar fashion. History of Kac-Moody algebras The initial construction by Elie Cartan and Wilhelm_Killing of finite dimensional simple Lie algebras from the Cartan integers was type dependent. In 1966 Jean-Pierre Serre showed that relations of Claude Chevalley and Harish-Chandra , with simplifications by N. Jacobson , give a defining presentation for the Lie algebra . One could thus describe a simple Lie algebra in terms of generators and relations using data from the matrix of Cartan integers, which is naturally positive definite. In his 1967 thesis, Robert Moody considered Lie algebras whose Cartan matrix is no longer positive definite . This still gave rise to a Lie algebra, but one which is now infinite dimensional. Simultaneously, Z-graded Lie algebras were being studied in Moscow where I. L. Kantor introduced and studied a general class of Lie algebras including what eventually became known as Kac-Moody algebras . Victor Kac was also studying simple or nearly simple Lie algebras with polynomial growth. A rich mathematical theory of infinite dimensional Lie algebras ro] evolved. An account of the subject, which also includes works of many others is given in (Kac 1990). See also (Seligman 1987) [9] . Definition A Kac-Moody algebra is given by the following: 1 . An nxn generalized Cartan matrix C = (c ..) of rank r. u 2. A vector space fjover the complex numbers of dimension 2« - r. 3. A set of n linearly independent elements OLi of fj and a set of n linearly independent elements a* of the dual space, such that a*{a.j) = C^„- . The CKi are known as coroots, while the a* are known as roots. The Kac-Moody algebra is the Lie algebra gdefined by generators &i and f^ and the elements of fjand relations [e;,/j] =0 for i^J [ei,x] = a*(x)ei, for x G f) [fi,x] = -Qi*(x)/ i ,forxG f) [x, x'] — Ofor iji'ef) ad{e i ) 1 -^(e j ) = Q Bd(/0 1 -°«(/i) = where ad : g — > End(g), ad(i)(y) = [x, y] is the adjoint representation of 0. A real (possibly infinite-dimensional) Lie algebra is also considered a Kac-Moody algebra if its complexification is a Kac-Moody algebra. Kac— Moody algebra 97 Interpretation fjis a Cartan subalgebra of the Kac— Moody algebra. If g is an element of the Kac— Moody algebra such that \/x e t), [g,x] =u{x)g where to is an element of fj*, then g is said to have weight m. The Kac— Moody algebra can be diagonalized into weight eigenvectors. The Cartan subalgebra h has weight zero, e. has weight a*, and/, has weight -a*.. If the Lie bracket of two weight eigenvectors is nonzero, then its weight is the sum of their weights. The condition [ e i> fj] — 0f° r * 7^ J s i m ply means the a*, are simple roots. Types of Kac-Moody algebras Properties of a Kac— Moody algebra are controlled by the algebraic properties of its generalized Cartan matrix C. In order to classify Kac— Moody algebras, it is enough to consider the case of an indecomposable matrixC, that is, assume that there is no decomposition of the set of indices / into a disjoint union of non-empty subsets / and / such that C .. = for all i in / and j in I . Any decomposition of the generalized Cartan matrix leads to the direct sum decomposition of the corresponding Kac— Moody algebra: where the two Kac— Moody algebras in the right hand side are associated with the submatrices of C corresponding to the index sets / and I . An important subclass of Kac— Moody algebras corresponds to symmetrizable generalized Cartan matrices C, which can be decomposed as DS, where D is a diagonal matrix with positive integer entries and S is a symmetric matrix. Under the assumptions that C is symmetrizable and indecomposable, the Kac— Moody algebras are divided into three classes: • A positive definite matrix S gives rise to a finite-dimensional simple Lie algebra. • A positive semidefinite matrix S gives rise to an infinite-dimensional Kac— Moody algebra of affine type, or an affine Lie algebra. • An indefinite matrix S gives rise to a Kac— Moody algebra of indefinite type. • Since the diagonal entries of C and S are positive, S cannot be negative definite or negative semidefinite. Symmetrizable indecomposable generalized Cartan matrices of finite and affine type have been completely classified. They correspond to Dynkin diagrams and affine Dynkin diagrams. Very little is known about the Kac— Moody algebras of indefinite type. Among those, the main focus has been on the (generalized) Kac— Moody algebras of hyperbolic type, for which the matrix S is indefinite, but for each proper subset of /, the corresponding submatrix is positive definite or positive semidefinite. Such matrices have rank at most 10 and have also been completely determined. Kac— Moody algebra Notes [1] (?) H. Garland and J. Lepowsky, Lie algebra homology and the Macdonald-Kac formulas, Invent. Math. 34 (1976), 37-76.] [2] Harish-Chandra On some applications of the universal enveloping algebra of a semisimple Lie algebra, Trans. Amer. Math. Soc. 70, (1951), 28--96 [3] Jacobson, N. Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10 Interscience Publishers (a division of John Wiley \& Sons), New York-London 1962 ix+331 pp. [4] Serre, J. -P., Serre, Algebres de Lie semi-simples complexes, (French) W. A. Benjamin, inc., New York-Amsterdam (1966) viii+130 pp. [5] Moody, R. V. , Lie algebras associated with generalized cartan matrices, Bull. Amer. Math. Soc, 73 (1967) 217-221 [6] Moody 1968, A new class of Lie algebras [7] Kantor, I. L. Graded Lie algebras (Russian) Trudy Sem. Vektor. Tenzor. Anal. 15 (1970), 227—266 [8] Kac, 1990 [9] Seligman, George B. Book Review: Infinite dimensional Lie algebras, Bull. Amer. Math. Soc. (N.S.) 16 (1987), no. 1, 144—149 [10] Carbone, L, Chung, S, Cobbs, C, McRae, R, Nandi, D, Naqvi, Y, and Penta, D, Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits, J. Phys. A: Math. Theor. 43 155209, 2010, arXiv: 1003.0564 (http://arxiv.org/abs/1003.0564) References • R.V. Moody, A new class of Lie algebras, J. of Algebra, 10 (1968) pp. 211-230 • V. Kac, Infinite dimensional Lie algebras, 3rd edition, Cambridge University Press (1990) ISBN 0521466938 (http://books. google. com/books?id=kuEjSb9teJwC&lpg=PPl&dq= Victor G.Kac&pg=PPl#v=onepage& q&f=false) • A. J. Wassermann, Lecture notes on Kac— Moody and Virasoro algebras (http://arxiv.org/abs/1004. 1287) • Hazewinkel, Michiel, ed. (2001), "Kac— Moody algebra" (http://eom.springer.de/Kyk055050.htm), Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104 • V.G. Kac, Simple irreducible graded Lie algebras of finite growth Math. USSR Izv., 2 (1968) pp. 1271—1311, Izv. Akad. Nauk USSR Ser. Mat., 32 (1968) pp. 1923-1967 External links • SIGMA: Special Issue on Kac-Moody Algebras and Applications (http://www.emis.de/journals/SIGMA/ Kac-Moody_algebras . html) Hopf algebra 99 Hopf algebra In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a (unital associative) algebra, a (counital coassociative) coalgebra, with these structures compatible making it a bialgebra, and moreover is equipped with an antiautomorphism satisfying a certain property. Hopf algebras occur naturally in algebraic topology, where they originated and are related to the H-space concept, in group scheme theory, in group theory (via the concept of a group ring), and in numerous other places, making them probably the most familiar type of bialgebra. Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems on the other. Formal definition Formally, a Hopf algebra is a bialgebra H over a field K together with a ^T-linear map S: H — > H (called the antipode) such that the following diagram commutes: S0id *H id(g>S Here A is the comultiplication of the bialgebra, V its multiplication, 11 its unit and e its counit. In the sumless Sweedler notation, this property can also be expressed as S(c(i))c( 2 ) = c (i)S( c (2)) — e(c)l for all c G H. As for algebras, one can replace the underlying field K with a commutative ring R in the above definition. The definition of Hopf algebra is self-dual (as reflected in the symmetry of the above diagram), so if one can define a dual of H (which is always possible if H is finite-dimensional), then it is automatically a Hopf algebra. Properties of the antipode The antipode S is sometimes required to have a ^-linear inverse, which is automatic in the finite-dimensional case, or if H is commutative or cocommutative (or more generally quasitriangular). In general, S is an antihomomorphism, so g-^is a homomorphism, which is therefore an automorphism if S was invertible (as may be required). If g 2 — j^d > men me Hopf algebra is said to be involutive (and the underlying algebra with involution is a *-algebra). If H is finite-dimensional semisimple over a field of characteristic zero, commutative, or cocommutative, then it is involutive. If a bialgebra B admits an antipode S, then S is unique ("a bialgebra admits at most 1 Hopf algebra structure"). The antipode is an analog to the inversion map on a group that sends Q to n~ . Hopf algebra 100 Hopf subalgebras A subalgebra K (not to be confused with the Field K in the notation above) of a Hopf algebra H is a Hopf subalgebra if it is a subcoalgebra of H and the antipode S maps K into K. In other words, a Hopf subalgebra K is a Hopf algebra in its own right when the multiplication, comultiplication, counit and antipode of H is restricted to K (and additionally the identity 1 is required to be in K). The Nichols-Zoeller Freeness theorem established (in 1989) that either natural K-module H is free of finite rank if H is finite dimensional: a generalization of Lagrange's theorem for subgroups. As a corollary of this and integral theory, a Hopf subalgebra of a semisimple finite dimensional Hopf algebra is automatically semisimple. A Hopf subalgebra K is said to be right normal in a Hopf algebra H if it satisfies the condition of stability, ad T (h)(K) C K foi all h in H, where the right adjoint mapping ad r is defined by ad T (h) (k) = S (h(i\)kh(2)ioT all k in K, h in H. Similarly, a Hopf subalgebra K is left normal in H if it is stable under the left adjoint mapping defined by adAhMk) = h(i\kS(hf2)) ■ The two conditions of normality are equivalent if the antipode S is bijective, in which case K is said to be a normal Hopf subalgebra. A normal Hopf subalgebra K in H satisfies the condition (of equality of subsets of H): JJJ£+ = K + H where ^"+denotes the kernel of the counit on K. This normality condition implies that JJ^+is a Hopf ideal of H (i.e. an algebra ideal in the kernel of the counit, a coalgebra coideal and stable under the antipode). As a consequence one has a quotient Hopf algebra H I HK +a && epimorphism J{ — ). HjK^H . a theory analogous to that of normal mi subgroups and quotient groups in group theory. Examples Group algebra. Suppose G is a group. The group algebra KG is a unital associative algebra over K. It turns into a Hopf algebra if we define • A : KG -^ KG ® KG by A(g) = g ® g for all g in G • e: KG ^K by £(g) = 1 for all g in G • S : KG -^ KG by S(g) = g _1 for all g in G. Functions on a finite group. Suppose now that G is a finite group. Then the set K of all functions from G to K with pointwise addition and multiplication is a unital associative algebra over K, and K ® a is naturally isomorphic to a x (for G infinite, a ® a is a proper subset of K x ). The set a becomes a Hopf algebra if we define • A : K° -^ K GxG by A(f)(x,y) =f(xy) for all /in K G and all x,y in G jC C • e : a — > K by e(/) =f(e) for every fin K [here e is the identity element of G] • S : K G -^ K G by S(f)(x) =f(x~ l ) for all/in K° and all x in G. Note that functions on a finite group can be identified with the group ring, though these are more naturally thought of as dual — the group ring consists of finite sums of elements, and thus pairs with functions on the group by evaluating the function on the summed elements. Regular functions on an algebraic group. Generalizing the previous example, we can use the same formulas to show that for a given algebraic group G over K, the set of all regular functions on G forms a Hopf algebra. Universal enveloping algebra. Suppose g is a Lie algebra over the field K and U is its universal enveloping algebra. U becomes a Hopf algebra if we define • A : U — > U ® t/ by A(x) =x®l+l®x for every x in g (this rule is compatible with commutators and can therefore be uniquely extended to all of U). • £ : U — > K by e(x) = for all x in g (again, extended to U) • S : U — > U by S(x) = -x for all x in g. Hopf algebra 101 Cohomology of Lie groups The cohomology algebra of a Lie group is a Hopf algebra: the multiplication is provided by the cup-product, and the comultiplication H*(G) ^H*(GxG) = H*(G) ® H*(G) by the group multiplication G X G — > G ■ This observation was actually a source of the notion of Hopf algebra. Using this structure, Hopf proved a structure theorem for the cohomology algebra of Lie groups. Theorem (Hopf) Let A be a finite-dimensional, graded commutative, graded cocommutative Hopf algebra over a field of characteristic 0. Then A (as an algebra) is a free exterior algebra with generators of odd degree. Quantum groups and non-commutative geometry All examples above are either commutative (i.e. the multiplication is commutative) or co-commutative (i.e. A = T o A where T: H <E> H — > H <E> H is defined by T(x <E> y) = y ® x). Other interesting Hopf algebras are certain "deformations" or "quantizations" of those from example 3 which are neither commutative nor co-commutative. These Hopf algebras are often called quantum groups, a term that is so far only loosely defined. They are important in noncommutative geometry, the idea being the following: a standard algebraic group is well described by its standard Hopf algebra of regular functions; we can then think of the deformed version of this Hopf algebra as describing a certain "non-standard" or "quantized" algebraic group (which is not an algebraic group at all). While there does not seem to be a direct way to define or manipulate these non-standard objects, one can still work with their Hopf algebras, and indeed one identifies them with their Hopf algebras. Hence the name "quantum group". Related concepts Graded Hopf algebras are often used in algebraic topology: they are the natural algebraic structure on the direct sum of all homology or cohomology groups of an H-space. Locally compact quantum groups generalize Hopf algebras and carry a topology. The algebra of all continuous functions on a Lie group is a locally compact quantum group. Quasi-Hopf algebras are generalizations of Hopf algebras, where coassociativity only holds up to a twist. Weak Hopf algebras, or quantum groupoids, are generalizations of Hopf algebras. Like Hopf algebras, weak Hopf algebras form a self-dual class of algebras; i.e., if H is a (weak) Hopf algebra, so is JJ* , the dual space of linear forms on H (with respect to the algebra-coalgebra structure obtained from the natural pairing with H and its coalgebra-algebra structure). A weak Hopf algebra H is usually taken to be a 1) finite dimensional algebra and coalgebra with coproduct A : i? — > H ® H an d counit e : H — > k satisfying all the axioms of Hopf algebra except possibly A(l) 7^ 1 ® l° r e(ab) ^ e(a)e(b)ior some a,b in H. Instead one requires that (A(l) ® 1)(1 ® A(l)) = (A(l) ® 1)(1 ® A{1)) = (A <g> Id)A(l)and e(abc) = ^2 € { a b(i))e{b(2) c ) = ^e(a&( 2 )) e (&(i) c ) for a11 a ' b ' andc inH - 2) H has a weakened antipode S : H — > H satisfying the axioms (a) - (c): (a) S(an\}a,(2) = lmefalrtOfor all a in H (the right-hand side is the interesting projection usually denoted by Jl R (a) oj: e s (a)with image a separable subalgebra denoted by jj R or H s )', (b) ar-\\S(a(2) = e(lma)l(2')f or a ^ a in H (another interesting projection usually denoted by Yl L (a) or: e^ (a) with image a separable algebra JJ L or JJt? anti -isomorphic to jj L via S); and (c) S{an'\)af2)S[a^\ — iS(a)for all a in H. Note that if A(l) = 1 ® 1, these conditions W^<s&l^^ e a^f J mVf ] c9mi^mmm feg SBte^f gf i^SMlm^S^ rigid tensor category. The unit H-module is the separable algebra fjL mentioned above. Hopf algebra 102 For example, a finite groupoid algebra is a weak Hopf algebra. In particular, the groupoid algebra on [n] with one pair of invertible arrows &ij and £ji between i and j in [n] is isomorphic to the algebra H of n x n matrices. The weak Hopf algebra structure on this particular H is given by coproduct A(e^) — e^- ® e^-, counit e(e;j) = 1 and antipode Sieij) = e™ . The separable subalgebras jj^and jj r coincide and are non-central commutative algebras in this particular case (the subalgebra of diagonal matrices). Early theoretical contributions to weak Hopf algebras are to be found in as well as Hopf algebroids introduced by J.-H. Lu in 1996 as a result on work on groupoids in Poisson geometry (later shown equivalent in nontrivial way to a construction of Takeuchi from the 1970s and another by Xu around the year 2000): Hopf algebroids generalize weak Hopf algebras and certain skew Hopf algebras. They may be loosely thought of as Hopf algebras over a noncommutative base ring, where weak Hopf algebras become Hopf algebras over a separable algebra. It is a theorem that a Hopf algebroid satisfying a finite projectivity condition over a separable algebra is a weak Hopf algebra, and conversely a weak Hopf algebra H is a Hopf algebroid over its separable subalgebra JJ L . The antipode axioms have been changed by G. Bohm and K. Szlachanyi (J. Algebra) in 2004 for tensor categorical reasons and to accommodate examples associated to depth two Frobenius algebra extensions. A left Hopf algebroid (H,R) is a left bialgebroid together with an antipode: the bialgebroid (H,R) consists of a total algebra H and a base algebra R and two mappings, an algebra homomorphism s : R — > H called a source map, an algebra anti-homomorphism t \ R > H called a target map, such that the commutativity condition s ( r l)t( r 2) = t( r 2) s ( r l)^ s satisfied for all 7"i, T^ £ R ■ The axioms resemble those of a Hopf algebra but are complicated by the possibility that R is a noncommutative algebra or its images under s and t are not in the center of H. In particular a left bialgebroid (H,R) has an R-R-bimodule structure on H which prefers the left side as follows: 7"i ■ h ■ T2 — s(ri)t{r2)h for all h in H, r^, r^ £ R ■ There is a coproduct A : H — > H ®# H and counit e \ H — > R that make (H, R, A, e)an R-coring (with axioms like that of a coalgebra such that all mappings are R-R-bimodule homomorphisms and all tensors over R). Additionally the bialgebroid (H,R) must satisfy A(a&) = A(a)A(6)for all a,b in H, and a condition to make sure this last condition makes sense: every image point A(a) satisfies an-*t(r) ® «(2) — Q(i) ® a( 2 )s(r)for all r in R. Also A(l) = 1 ® 1 • The counit is m^S&faMgSs r^"-\l?V a t?s d uafy CC ^ V(l#)ar lfel|(^)avno&(o^m^)latisfymg conditions of exchanging the source and target maps and satisfying two axioms like Hopf algebra antipode axioms; see the references in Lu or in Bohm-Szlachanyi for a more example-category friendly, though somewhat more complicated, set of axioms for the antipode S. The latter set of axioms depend on the axioms of a right bialgebroid as well, which are a straightforward switching of left to right, s with t, of the axioms for a left bialgebroid given above. As an example of left bialgebroid, take R to be any algebra over a field k. Let H be its algebra of linear self-mappings. Let s(r) be left multication by r on R; let t(r) be right multiplication by r on R. H is a left bialgebroid over R, which may be seen as follows. From the fact that H ®r H = Honifefi? (x) i? 5 i?)one may define a coproduct by A(/)(r®ii) = f(ru)ior each linear transformation f from R to itself and all r,u in R. Coassociativity of the coproduct follows from associativity of the product on R. A counit is given by e(/) = f(l) . The counit axioms of a coring follow from the identity element condition on multiplication in R. The reader will be amused, or at least edified, to check that (H,R) is a left bialgebroid. In case R is an Azumaya algebra, in which case H is isomorphic to R tensor R, an antipode comes from transposing tensors, which makes H a Hopf algebroid over R. Multiplier Hopf algebras introduced by Alfons Van Daele in 1994 are generalizations of Hopf algebras where comultiplication from an algebra (with or withthout unit) to the multiplier algebra of tensor product algebra of the algebra with itself. Hopf group-(co)algebras introduced by V.G.Turaev in 2000 are also generalizations of Hopf algebras. Hopf algebra 103 Analogy with groups Groups can be axiomatized by the same diagrams (equivalently, operations) as a Hopf algebra, where G is taken to be a set instead of a module. In this case: • the field K is replaced by the 1 -point set • there is a natural counit (map to 1 point) • there is a natural comultiplication (the diagonal map) • the unit is the identity element of the group • the multiplication is the multiplication in the group • the antipode is the inverse In this philosophy, a group can be thought of as a Hopf algebra over the "field with one element". See also • Quasitriangular Hopf algebra • Algebra/set analogy • Representation theory of Hopf algebras • Ribbon Hopf algebra • Superalgebra • Supergroup • Anyonic Lie algebra Notes [1] Dascalescu, Nastasescu & Raianu (2001), Prop. 4.2.6, p. 153 (http://books.google.com/books?id=pBJ6sbPHA0IC&pg=PA153&dq="is+ an+antimorphism+of+ algebras ") [2] Dascalescu, Nastasescu & Raianu (2001), Remarks 4.2.3, p. 151 (http://books.google.com/books ?id=pBJ6sbPHA0IC&pg=PA151& dq="the+antipode+is+unique") [3] Quantum groups lecture notes (http://www.mathematik.uni-muenchen.de/~pareigis/Vorlesungen/98SS/Quantum_Groups/LN2_l.PDF) [4] S. Montgomery, Hopf algebras and their actions on rings, Conf. Board in Math. Sci. vol. 82, A.M.S., 1993. ISBN 0-8218-0738-2 [5] Hopf, 1941. [6] Gabriella Bohm, Florian Nill, Kornel Szlachanyi. J. Algebra 221 (1999), 385-438 [7] Dmitri Nikshych, Leonid Vainerman, in: New direction in Hopf algebras, S. Montgomery and H.-J. Schneider, eds., M.S. R.I. Publications, vol. 43, Cambridge, 2002, 211-262. [8] Alfons Van Daele. Multiplier Hopf algebras (http://www.ams.org/tran/1994-342-02/S0002-9947-1994-1220906-5/ S0002-9947-1994-1220906-5.pdf), Transactions of the American Mathematical Society 342(2) (1994) 917-932 [9] Group = Hopf algebra « Secret Blogging Seminar (http://sbseminar.wordpress.com/2007/10/07/group-hopf-algebra/), Group objects and Hopf algebras (http://www.youtube.com/watch?v=p3kkm5dYH-w), video of Simon Willerton. References • Dascalescu, Sorin; Nastasescu, Constantin; Raianu, §erban (2001), Hopf Algebras, Pure and Applied Mathematics, 235 (1st ed.), Marcel Dekker, ISBN 0-8247-0481-9. • Pierre Carrier, A primer of Hopf algebras (http://inc.web.ihes.fr/prepub/PREPRINTS/2006/M/M-06-40. pdf), IHES preprint, September 2006, 81 pages • Jurgen Fuchs, Affine Lie Algebras and Quantum Groups, (1992), Cambridge University Press. ISBN 0-521 -484 12-X • H. Hopf, Uber die Topologie der Gruppen-Mannigfaltigkeiten und ihrer Verallgemeinerungen, Ann. of Math. 42 (1941), 22-52. Reprinted in Selecta Heinz Hopf, pp. 119-151, Springer, Berlin (1964). MR4784 • Street, Ross (2007), Quantum groups, Australian Mathematical Society Lecture Series, 19, Cambridge University Press, MR2294803, ISBN 978-0-521-69524-4; 978-0-521-69524-4. Quantum group 104 Quantum group In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra. There is no single, all-encompassing definition, but instead a family of broadly similar objects. The term "quantum group" often denotes a kind of noncommutative algebra with additional structure that first appeared in the theory of quantum integrable systems, and which was then formalized by Vladimir Drinfel'd and Michio Jimbo as a particular class of Hopf algebra. The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a "bicrossproduct' class of quantum groups introduced by Shahn Majid a little after the work of Drinfeld and Jimbo. In Drinfeld's approach, quantum groups arise as Hopf algebras depending on an auxiliary parameter q or h, which become universal enveloping algebras of a certain Lie algebra, frequently semisimple or affine, when q = 1 or h = 0. Closely related are certain dual objects, also Hopf algebras and also called quantum groups, deforming the algebra of functions on the corresponding semisimple algebraic group or a compact Lie group. Just as groups often appear as symmetries, quantum groups act on many other mathematical objects and it has become fashionable to introduce the adjective quantum in such cases; for example there are quantum planes and quantum Grassmannians. Intuitive meaning The discovery of quantum groups was quite unexpected, since it was known for a long time that compact groups and semisimple Lie algebras are "rigid" objects, in other words, they cannot be "deformed". One of the ideas behind quantum groups is that if we consider a structure that is in a sense equivalent but larger, namely a group algebra or a universal enveloping algebra, then a group or enveloping algebra can be "deformed", although the deformation will no longer remain a group or enveloping algebra. More precisely, deformation can be accomplished within the category of Hopf algebras that are not required to be either commutative or cocommutative. One can think of the deformed object as an algebra of functions on a "noncommutative space", in the spirit of the noncommutative geometry of Alain Connes. This intuition, however, came after particular classes of quantum groups had already proved their usefulness in the study of the quantum Yang-Baxter equation and quantum inverse scattering method developed by the Leningrad School (Ludwig Faddeev, Leon Takhtajan, Evgenii Sklyanin, Nicolai Reshetikhin and Korepin) and related work by the Japanese School. The intuition behind the second, bicrossproduct, class of quantum groups was different and came from the search for self-dual objects as an approach to quantum gravity . Drinfel'd-Jimbo type quantum groups One type of objects commonly called a "quantum group" appeared in the work of Vladimir Drinfel'd and Michio Jimbo as a deformation of the universal enveloping algebra of a semisimple Lie algebra or, more generally, a Kac-Moody algebra, in the category of Hopf algebras. The resulting algebra has additional structure, making it into a quasitriangular Hopf algebra. Let A = (a^-)be the Cartan matrix of the Kac-Moody algebra, and let q be a nonzero complex number distinct from 1, then the quantum group, U„(G), where G is the Lie algebra whose Cartan matrix is A, is defined as the unital associative algebra with generators k\ (where \ is an element of the weight lattice, i.e. 2(A. aA /(cti, Qtj) G Z for all /), and &i and f+ (for simple roots, Q^), subject to the following relations: * fa = 1, * k\k^ — k\ +fl , Quantum group 105 k — k~ x . r e . f.i =s-.- -i-, ft -ft 1 ~ aij U — n 1 I • V (-l) n - jki ' e n ee~ aii ~ n = 0,f°r i^i, 1_a 'J [1 _ n 1 I n where ki — k ai , g. — na^^), [0]gj = 1' [ n ]gJ = ii [ m ]gif or a ^ positive integers n, and m=l q™ _ q -™ [m] Q . = — ^j— .These are the q-factorial and q-number, respectively, the q-analogs of the ordinary factorial. ft - % The last two relations above are the g-Serre relations, the deformations of the Serre relations. In the limit as q — > 1, these relations approach the relations for the universal enveloping algebra [/(G), where k\ — k_\ k x — y 1 and 5- t x as q — > 1 , where the element, £ A , of the Cartan subalgebra satisfies q - q' 1 (t^^ h) = Mh) for all h in the Cartan subalgebra. There are various coassociative coproducts under which these algebras are Hopf algebras, for example, • Ax(fc A ) = k x ®k x , Aife) = 1 ® e, + e^ ® fc, , Ai(/0 = fc" 1 ®fi + fi®L • A 2 (fc A ) = k x ® k x , A 2 (ei) = k' 1 <g> a + e { ® 1, A 2 (/,) = 1 ® £ + /; ® ft* . • A 3 (ft A ) = ft* ® A: A , A 3 (ei) = fc~* ® e» + e, ® A:?. A 3 (/0 = A,"' ® /; + /; ® A:?- where the set of generators has been extended, if required, to include k x for X which is expressible as the sum of an element of the weight lattice and half an element of the root lattice. In addition, any Hopf algebra leads to another with reversed copproduct T"oA> where J 1 is given by T(x ® y) = y ® X, giving three more possible versions. The counit on f/„(j4)is the same for all these coproducts: e(ft A ) = 1, e(e^) = 0, t(fi) = 0, and the respective antipodes for the above coproducts are given by • 5i(fc A ) = ft_ A , Si(ei) = -eifcf 1 , Si(/0 = -fci/i. • 5 2 (A:a) = &-*, 5 2 (ei) = -ft;e { , £ 2 (/i) = -fik' 1 , ' S 3 (k x ) = ft_ A , S 3 (ei) = -fte*, SsCfi) = -q~ X fi. Alternatively, the quantum group U„(G) can be regarded as an algebra over the field C(q), the field of all rational functions of an indeterminate q over (£ . Similarly, the quantum group [/„(£?) can be regarded as an algebra over the field Q(g), the field of all rational functions of an indeterminate q over Q (see below in the section on quantum groups at q = 0). The center of quantum group can be described by quantum determinant. Representation theory Just as there are many different types of representations for Kac-Moody algebras and their universal enveloping algebras, so there are many different types of representation for quantum groups. As is the case for all Hopf algebras, U„ (G) has an adjoint representation on itself as a module, with the action being given by Ad ^ = Z^U^O^)). where A ( x ) = E x (l) ® *(2). (x) (*) Quantum group 106 Case 1: q is not a root of unity One important type of representation is a weight representation, and the corresponding module is called a weight module. A weight module is a module with a basis of weight vectors. A weight vector is a nonzero vector v such that k\.V = d\i;for all \ , where d\ are complex numbers for all weights \ such that • d = l, • dxdp = d\ +jl , for all weights \ and /i . A weight module is called integrable if the actions of &i and f^ are locally nilpotent (i.e. for any vector v in the module, there exists a positive integer k, possibly dependent on v, such that e^ .v = f^.v = Ofor all i). In the case of integrable modules, the complex numbers d\ associated with a weight vector satisfy d , = c\q^' v ^ > where f is an element of the weight lattice, and C\ are complex numbers such that * Pd = 1, • c X c /i = c A+/i, for all weights _\ and jJ. , * C 2q . = 1 for all/. Of special interest are highest weight representations, and the corresponding highest weight modules. A highest weight module is a module generated by a weight vector v, subject to k\.V = d\i;for all weights \ , and e{.V = Ofor all i. Similarly, a quantum group can have a lowest weight representation and lowest weight module, i.e. a module generated by a weight vector v, subject to k\.V = d\i;for all weights \ , and f^.v = Ofor all i. Define a vector v to have weight i/if k\.V = q^'^v f° r all A m tne weight lattice. If G is a Kac-Moody algebra, then in any irreducible highest weight representation of U„(G) , with highest weight V , the multiplicities of the weights are equal to their multiplicities in an irreducible representation of U(G) with equal highest weight. If the highest weight is dominant and integral (a weight jJ- is dominant and integral if jJ- satisfies the condition that 2(u, ceA/fai, CtAis a non-negative integer for all i), then the weight spectrum of the irreducible representation is invariant under the Weyl group for G, and the representation is integrable. Conversely, if a highest weight module is integrable, then its highest weight vector v satisfies k\.v = C\q^' v ^v > where C\ are complex numbers such that * Co = 1. • c \ c fi = c X+fj,, for all weights \ and (l , * c 2a f = 1 for all i, and yis dominant and integral. As is the case for all Hopf algebras, the tensor product of two modules is another module. For an element x of U„{G), and for vectors v and w in the respective modules, x.iv ® w) = £±{x).{v ® w), so that k x .(v (g> w) = k\.v ® k x .w , and in the case of coproduct Ai, e^.(^ <g> w) = ki.v ® e^.u* + ej.u ® w and /..(u (g) w ) = v g) /^.u; + /..?; <g> fcr 1 .^ . The integrable highest weight module described above is a tensor product of a one-dimensional module (on which k x = C\ for all \ , and e^ = /^ = Ofor all /) and a highest weight module generated by a nonzero vector l>o, subject to k\.VQ = q^^v^ * all weights \ , and e{.VQ = Ofor all j. In the specific case where G is a finite-dimensional Lie algebra (as a special case of a Kac-Moody algebra), then the irreducible representations with dominant integral highest weights are also finite-dimensional. In the case of a tensor product of highest weight modules, its decomposition into submodules is the same as for the tensor product of the corresponding modules of the Kac-Moody algebra (the highest weights are the same, as are their multiplicities). Quantum group 107 Quasitriangularity Case 1: q is not a root of unity Strictly, the quantum group U„(G) is not quasitriangular, but it can be thought of as being "nearly quasitriangular" in that there exists an infinite formal sum which plays the role of an 7?-matrix. This infinite formal sum is expressible in terms of generators &i and f^ , and Cartan generators t x , where k x is formally identified with «** . The infinite formal sum is the product of two factors, n^^j tx 3® t,J, j, and an infinite formal sum, where {A,} is a basis for the dual space to the Cartan subalgebra, and {uA is the dual basis, and 77 is a sign (+1 or -1). The formal infinite sum which plays the part of the K-matrix has a well-defined action on the tensor product of two irreducible highest weight modules, and also on the tensor product if two lowest weight modules. Specifically, if v has weight a and w has weight j3 , then ^z^j^j^^j (v ® w) = n^^'^v £R> w > an ^ tne ^ act tnat tne modules are both highest weight modules or both lowest weight modules reduces the action of the other factor on V (x) W to a finite sum. Specifically, if V is a highest weight module, then the formal infinite sum, R, has a well-defined, and invertible, action on V ® V> an d this value of R (as an element of Hom(T/) ® Hom(l/)) satisfies the Yang-Baxter equation, and therefore allows us to determine a representation of the braid group, and to define quasi-invariants for knots, links and braids. Quantum groups at q = Masaki Kashiwara has researched the limiting behaviour of quantum groups as q — 5- . As a consequence of the defining relations for the quantum group U„{G), U„{G) can be regarded as a Hopf algebra over Q(g), the field of all rational functions of an indeterminate q over Q . For simple root Q^and non-negative integer n, define e - ' = e™/\n] . ! a nd fy 1 ' = f™/\n] .'(specifically, e- = f- = 1)- in an integrable module ]\/[ , and for weight _\ , a vector u £ M x (i-e. a vector liin M with weight \ ) can be uniquely decomposed into the sums oo oo /(n) V^ (n) where u n G ker( ei ) H M A+TlQ . , v n G ker(/ 4 ) D M x _ nOLi , u n ^ Oonly if n+ -f ^ > 0, and v n ^ Oonly if n — - — - — ^- > 0. Linear mappings e^ : M — > M and f . ■ M — *• M can be defined on M A by oo V"^ r(n-l) V"^ (ti+1) • e { U = 2^ fi Un = 2_s e i V ™ Ti=l Ti=0 OC OC 7i.=0 71=1 Let ^4 be the integral domain of all rational functions in Q(g) which are regular at q = (/.<?. a rational function /(q)is an element of ^4 if and only if there exist polynomials g(q) and h(q) in the polynomial ring Q[g] such that /l(0) 7^ 0, and f(q) — g(q)/h(q))- A crystal base for jl/f is an ordered pair (L, 5), such that • £j is a free ^4 -submodule of J^f such that M. = Qfo) ®A L] • £? is a Q -basis of the vector space L/qL over (Q), • L = ® X L X and B = U X B X , where L x = L D M A and £ A = 5 H (L x /qL x ) y • e { L C L and /.£, ^ L for all z, Quantum group 108 • CiB C B U {0} and / ; BcBU {0} for all i, * for all b € B and &' (E 5, and for all z, e,b = b' if and only if fib' — 6. To put this into a more informal setting, the actions of e^ft and /^ej are generally singular at q = Oon an integrable module M . The linear mappings e^and t . on the module are introduced so that the actions of g.f. and f.g.are regular at q = Oon the module. There exists a Q(qA -basis of weight vectors £j for J\^f , with respect to which the actions of e^ and f. are regular at q = Ofor all /. The module is then restricted to the free A -module generated by the basis, and the basis vectors, the A -submodule and the actions of e^ and f. are evaluated at q = . Furthermore, the basis can be chosen such that at q = , for all j , e^ and f. are represented by m^mm^t^^¥MlM%Ti^^^W^ X ^^^^^ edges. Each vertex of the graph represents an element of the Q -basis £?of LlqL , and a directed edge, labelled by i, and directed from vertex fito vertex V2, represents that fy, = f.K(and, equivalently, that b\ = ej&2)> where ft^is the basis element represented by 1*1, and fe 2 i s the basis element represented by Vi. The graph completely determines the actions of e^and f. at q = 0. If an integrable module has a crystal base, then the module is irreducible if and only if the graph representing the crystal base is connected (a graph is called "connected" if the set of vertices cannot be partitioned into the union of nontrivial disjoint subsets 1/jand V2 suc h that there are no edges joining any vertex in Vjto any vertex in V^)- For any integrable module with a crystal base, the weight spectrum for the crystal base is the same as the weight spectrum for the module, and therefore the weight spectrum for the crystal base is the same as the weight spectrum for the corresponding module of the appropriate Kac -Moody algebra. The multiplicities of the weights in the crystal base are also the same as their multiplicities in the corresponding module of the appropriate Kac-Moody algebra. It is a theorem of Kashiwara that every integrable highest weight module has a crystal base. Similarly, every integrable lowest weight module has a crystal base. Tensor products of crystal bases Let Mbt an integrable module with crystal base (L, S)and M'be an integrable module with crystal base (L',B')- For crystal bases, the coproduct /\, given by A(fc A ) = k x ® k x , A(e;) = e; ® k' 1 + 1 ® e h A(/;) = f t ® 1 + k { ® ft , is adopted. The integrable module M ®Q( g ) M' has crystal base (£, & A L' \B ® B'), where B ® B' - {b ®qb' : b G B, b' £ B'} ■ For a basis vector b G B, define £i(6) = max{n > : e"fo / 0} and 0.(&) = max {n > : ^6 ^ 0} • The actions of g-and /.on 6 ?' gi ( 6 « 6 -)- /8i6 ® i/ ' if * (6) - £i(6,) ' b®eiV, if 0i(6) < e i (6') ) . ~f(h6*h>\-V ih ® h 'i $ Mb) > ^(V), M&W&J 16®/^, if <fc(&) < *(*). The decomposition of the product two integrable highest weight modules into irreducible submodules is determined by the decomposition of the graph of the crystal base into its connected components (i.e. the highest weights of the submodules are determined, and the multiplicity of each highest weight is determined). Quantum group 109 Compact matrix quantum groups See also compact quantum group. S.L. Woronowicz introduced compact matrix quantum groups. Compact matrix quantum groups are abstract structures on which the "continuous functions" on the structure are given by elements of a C*-algebra. The geometry of a compact matrix quantum group is a special case of a noncommutative geometry. The continuous complex-valued functions on a compact Hausdorff topological space form a commutative C*-algebra. By the Gelfand theorem, a commutative C*-algebra is isomorphic to the C*-algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C*-algebra up to homeomorphism. For a compact topological group, G, there exists a C*-algebra homomorphism A : C(G) — > C[G) ® C(G) (where C{G) ® CfGQis the C*-algebra tensor product - the completion of the algebraic tensor product of C(G) and C(G)), such that A(/)(x, y) = f(xy)foT all / £ C{G), and for all x, y G G (where {_/" <g) g)(x, y) — f(x)g(y) for all /, g G C(G)and all x, y £ G). There also exists a linear multiplicative mapping K ; C(G) -> C{G), such that k(/)(x) = /(x _1 )for all / £ C(G)and all x £ G ■ Strictly, this does not make C(G)a Hopf algebra, unless G is finite. On the other hand, a finite-dimensional representation of G can be used to generate a *-subalgebra of C(G) which is also a Hopf *-algebra. Specifically, if g \— > (itj.-f^lL _■ is an n -dimensional representation of Q , then u^j £ C(G)for all i, j ' , and ^H^ii/ = ^__, u ik ® ^fcjfor all fc ■i,_7 . It follows that the *-algebra generated by ^ijfor all i 7 j and K(itj,)for all i,j is a Hopf *-algebra: the counit is determined by e(u^) = S^j for all i,^ (where ^jis the Kronecker delta), the antipode is k, and the ffi^^Sh^rSlfeation, aiempSct^mSmx qu&ftium group Is 'defined as a pair (C it), where (J is a C*-algebra and ti — (uij)i j—i u is a matrix with entries in (7 such that • The *-subalgebra, Cq, of (J , which is generated by the matrix elements of u , is dense in (J ; • There exists a C*-algebra homomorphism A : C — > G ® C (where (J (g (J is the C*-algebra tensor product - the completion of the algebraic tensor product of (J and (J ) such that ^\ u ij ) = 2^/ Uik Uk i for all i,j( A is called the comultiplication); • There exists a linear antimultiplicative map k, : Cq — ► Co( me coinverse) such that k(k,(v*)*) = u for all V G Co a nd / j K \ u ik) u kj = / s U ik K \ u kj) = O-ij-'jwhere /is the identity element of (J . Since Kis fe fe antimultiplicative, then k(ui(;) = K(w)K,(v)for all i>,ii; £ Co- As a consequence of continuity, the comultiplication on (J is coassociative. In general, (J is not a bialgebra, and Cois a Hopf *-algebra. Informally, (7 can t> e regarded as the *-algebra of continuous complex-valued functions over the compact matrix quantum group, and u can be regarded as a finite-dimensional representation of the compact matrix quantum group. A representation of the compact matrix quantum group is given by a corepresentation of the Hopf *-algebra (a corepresentation of a counital coassociative coalgebra ^4 is a square matrix v = (ifoji 7 =i n w i m entries in A n (so v G M n (A)) such that A(u^-) = 2J v ik ® w^-for all i 7 j and e(t>ij) — S^fov all i 7 j ). Furthermore, fc=l a representation v, is called unitary if the matrix for v is unitary (or equivalently, if k(i>;j) = v^ for all i,j). An example of a compact matrix quantum group is SUAty, where the parameter /i is a positive real number. So 5^(2) - (C(5T/ At (2), u), where C(S , f7 /i (2))is the C*-algebra generated by a and 7,subjectto 77* = 7*7i a l = PI 01 -! a l* = / i 7* ct 5 aa * + ^7*7 = a * a + /^ _ 7*7 = I> Quantum group 110 and 7i = I ^ * 1 , so that the comultiplication is determined by A (a) = aCS>a; — 7®7*> A(7) = a ® 7 + 7 ® a", and the coinverse is determined by k(o) — a* , ^(7) = — /i _1 7» ^(7*) = —U"f* > /c(ct*) = a • Note that ii is a representation, but not a unitary representation, u is equivalent to the unitary representation v = I 1 * * J ■ Equivalently, S 1/^(2) = ((7(5^(2)), u;), where C(5f/ /i (2))is the C*-algebra generated by ct and (3 .subject to (3/3* = p*p, a/3 = p/3a, aft* = (ift*a, aa* + p?ft*ft = a*a + ft*ft = I, and w = ( Q ^ * ) : so triat me comultiplication is determined by A (a) = a ® ct — Lif3 ® [3* , A(/5) = a ® f3 + /3® a*, and the coinverse is determined by k(q) = a*, «;(/?) = —/i _1 /3, k(/3*) = —il(3*, /c(a*) = a- Note that ii;is a unitary representation. The realizations can be identified by equating 7 = ^/^ . When /i = 1, then 5'f/„(2)is equal to the algebra C(SU(2))of functions on the concrete compact group SU(2). Bicrossproduct quantum groups Whereas compact matrix pseudogroups are typically versions of Drinfeld-Jimbo quantum groups in a dual function algebra formulation, with additional structure, the bicrossproduct ones are a distinct second family of quantum groups of increasing importance as deformations of solvable rather than semisimple Lie groups. They are associated to Lie splittings of Lie algebras or local factorisations of Lie groups and can be viewed as the cross product or Mackey quantisation of one of the factors acting on the other for the algebra and a similar story for the coproduct /\ with the second factor acting back on the first. The very simplest nontrivial example corresponds to two copies of JJ locally acting on each other and results in a quantum group (given here in an algebraic form) with generators p K K~^, say, and coproduct \p,K\ =hK{K-l), Ap = p®K + l®p,AK = K®K where fa is the deformation parameter. This quantum group was linked to a toy model of Planck scale physics implementing Born reciprocity when viewed as a deformation of the Heisenberg algebra of quantum mechanics. Also, starting with any compact real form of a semisimple Lie algebra Q its complexification as a real Lie algebra of twice the dimension splits into Q and a certain solvable Lie algebra (the Iwasawa decomposition), and this provides a canonical bicrossproduct quantum group associated to Q . For su(2) one obtains a quantum group deformation of the Euclidean group E(3) of motions in 3 dimensions. Notes [1] Schwiebert, Christian (1994), Generalized quantum inverse scattering, arXiv:hep-th/9412237v3 [2] Majid, Shahn (1988), "Hopf algebras for physics at the Planck scale", Classical and Quantum Gravity 5: 1587—1607, doi: 10.1088/0264-9381/5/12/010 References • Podles, P.; Muller, E., Introduction to quantum groups, arXiv:q-alg/9704002 • Kassel, Christian (1995), Quantum groups, Graduate Texts in Mathematics, 155, Berlin, New York: Springer- Verlag, MR1321145, ISBN 978-0-387-94370-1 • Majid, Shahn (2002), A quantum groups primer, London Mathematical Society Lecture Note Series, 292, Cambridge University Press, MR1904789, ISBN 978-0-521-01041-2 Quantum group 111 • Street, Ross (2007), Quantum groups, Australian Mathematical Society Lecture Series, 19, Cambridge University Press, MR2294803, ISBN 978-0-521-69524-4; 978-0-521-69524-4. • Majid, Shahn (January 2006), "What Is. ..a Quantum Group?" (http://www.ams.org/notices/200601/what-is. pdf) (PDF), Notices of the American Mathematical Society 53 (1): 30—31, retrieved 2008-01-16 • Shnider, Steven; Sternberg, Shlomo (1993) Quantum groups. From coalgebras to Drinfel'd algebras. A guided tour. Graduate Texts in Mathematical Physics, II. International Press, Cambridge, MA. Affine quantum group In mathematics, a quantum affine algebra (or affine quantum group) is a Hopf algebra that is a ^-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced independently by Drinfeld (1985) and Jimbo (1985) as a special case of their general construction of a quantum group from a Cartan matrix. One of their principal applications has been to the theory of solvable lattice models in quantum statistical mechanics, where the Yang-Baxter equation occurs with a spectral parameter. Combinatorial aspects of the representation theory of quantum affine algebras can be described simply using crystal bases, which correspond to the degenerate case when the deformation parameter q vanishes and the Hamiltonian of the associated lattice model can be explicitly diagonalized. References • Drinfeld, V. G (1985), "Hopf algebras and the quantum Yang-Baxter equation", Doklady Akademii Nauk SSSR 283 (5): 1060-1064, MR802128, ISSN 0002-3264 • Drinfeld, V. G (1987), "A new realization of Yangians and of quantum affine algebras", Doklady Akademii Nauk SSSR 296 (1): 13-17, MR914215, ISSN 0002-3264 • Frenkel, Igor B.; Reshetikhin, N. Yu. (1992), "Quantum affine algebras and holonomic difference equations" , Communications in Mathematical Physics 146 (1): 1-60, doi:10.1007/BF02099206, MR1163666, ISSN 0010-3616 • Jimbo, Michio (1985), "A q-difference analogue of U(g) and the Yang-Baxter equation", Letters in Mathematical Physics. 10 (1): 63-69, doi:10.1007/BF00704588, MR797001, ISSN 0377-9017 • Jimbo, Michio; Miwa, Tetsuji (1995), Algebraic analysis of solvable lattice models, CBMS Regional Conference Series in Mathematics, 85, Published for the Conference Board of the Mathematical Sciences, Washington, DC, MR1308712, ISBN 978-0-8218-0320-2 References [1] http://projecteuclid.org/getRecord?id=euclid.cmp/ 1104249974 Group representation 112 Group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well-understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system. The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation. Some people use realization for the general notion and reserve the term representation for the special case of linear representations. The bulk of this article describes linear representation theory; see the last section for generalizations. Branches of group representation theory The representation theory of groups divides into subtheories depending on the kind of group being represented. The various theories are quite different in detail, though some basic definitions and concepts are similar. The most important divisions are: • Finite groups — Group representations are a very important tool in the study of finite groups. They also arise in the applications of finite group theory to crystallography and to geometry. If the field of scalars of the vector space has characteristic p, and if p divides the order of the group, then this is called modular representation theory; this special case has very different properties. See Representation theory of finite groups. • Compact groups or locally compact groups — Many of the results of finite group representation theory are proved by averaging over the group. These proofs can be carried over to infinite groups by replacement of the average with an integral, provided that an acceptable notion of integral can be defined. This can be done for locally compact groups, using Haar measure. The resulting theory is a central part of harmonic analysis. The Pontryagin duality describes the theory for commutative groups, as a generalised Fourier transform. See also: Peter-Weyl theorem. • Lie groups — Many important Lie groups are compact, so the results of compact representation theory apply to them. Other techniques specific to Lie groups are used as well. Most of the groups important in physics and chemistry are Lie groups, and their representation theory is crucial to the application of group theory in those fields. See Representations of Lie groups and Representations of Lie algebras. • Linear algebraic groups (or more generally affine group schemes) — These are the analogues of Lie groups, but over more general fields than just R or C. Although linear algebraic groups have a classification that is very similar to that of Lie groups, and give rise to the same families of Lie algebras, their representations are rather different (and much less well understood). The analytic techniques used for studying Lie groups must be replaced by techniques from algebraic geometry, where the relatively weak Zariski topology causes many technical complications. • Non-compact topological groups — The class of non-compact groups is too broad to construct any general representation theory, but specific special cases have been studied, sometimes using ad hoc techniques. The semisimple Lie groups have a deep theory, building on the compact case. The complementary solvable Lie groups cannot in the same way be classified. The general theory for Lie groups deals with semidirect products of the two types, by means of general results called Mackey theory, which is a generalization of Wigner's classification methods. Group representation 113 Representation theory also depends heavily on the type of vector space on which the group acts. One distinguishes between finite-dimensional representations and infinite-dimensional ones. In the infinite-dimensional case, additional structures are important (e.g. whether or not the space is a Hilbert space, Banach space, etc.). One must also consider the type of field over which the vector space is defined. The most important case is the field of complex numbers. The other important cases are the field of real numbers, finite fields, and fields of p-adic numbers. In general, algebraically closed fields are easier to handle than non-algebraically closed ones. The characteristic of the field is also significant; many theorems for finite groups depend on the characteristic of the field not dividing the order of the group. Definitions A representation of a group G on a vector space V over a field K is a group homomorphism from G to GL(V), the general linear group on V. That is, a representation is a map p: G -> GL(V) such that P(9i92) = P(9i)p(92), for all g 1 ,g 2 ^ G. Here V is called the representation space and the dimension of V is called the dimension of the representation. It is common practice to refer to V itself as the representation when the homomorphism is clear from the context. In the case where V is of finite dimension n it is common to choose a basis for V and identify GL(V) with GL («, K) the group of n-by-n invertible matrices on the field K. If G is a topological group and V is a topological vector space, a continuous representation of G on V is a representation p such that the application $ ; Q x V — > ^defined by <&(g, v) = p(g).V is continuous. The kernel of a representation pof a group G is defined as the normal subgroup of G whose image under pis the identity transformation: kery9= {g G G \ p(g) = id} . A faithful representation is one in which the homomorphism G — » GL(V) is injective; in other words, one whose kernel is the trivial subgroup {e} consisting of just the group's identity element. Given two K vector spaces V and W, two representations Pl :G^GL{V) and p 2 :G^GL(W) are said to be equivalent or isomorphic if there exists a vector space isomorphism a:V^W so that for all g in G a°pi{g) oo: -1 = P2(g)- Group representation 114 Examples Consider the complex number u = e which has the property u =1. The cyclic group C = 2 representation p on C given by: 1, u, u } has a PO) 1 1 p(u) 1 u ,(«•) This representation is faithful because p is a one-to-one map. An isomorphic representation for C is PQ) 1 1 p(u) u 1 M 1 u 2 u 2 1 The group C may also be faithfully represented on R by Ml) 1 1 p(u) a —b b a («•) a b —b a where a = ft(u) = -1/2 and & = $(„) = ^2- Reducibility A subspace W of V that is invariant under the group action is called a subrepresentation. If V has exactly two subrepresentations, namely the zero-dimensional subspace and V itself, then the representation is said to be irreducible; if it has a proper subrepresentation of nonzero dimension, the representation is said to be reducible. The representation of dimension zero is considered to be neither reducible nor irreducible, just like the number 1 is considered to be neither composite nor prime. Under the assumption that the characteristic of the field K does not divide the size of the group, representations of finite groups can be decomposed into a direct sum of irreducible subrepresentations (see Maschke's theorem). This holds in particular for any representation of a finite group over the complex numbers, since the characteristic of the complex numbers is zero, which never divides the size of a group. In the example above, the first two representations given are both decomposable into two 1 -dimensional subrepresentations (given by span{(l,0)} and span{(0,l)}), while the third representation is irreducible. Generalizations Set-theoretical representations A set-theoretic representation (also known as a group action or permutation representation) of a group G on a set X x is given by a function p from G to X , the set of functions from X to X, such that for all g , g in G and all x in X: p(l)[x] =x P(9i92)[x] = P(9i)[p(92)[x]}. This condition and the axioms for a group imply that p(g) is a bijection (or permutation) for all g in G. Thus we may equivalently define a permutation representation to be a group homomorphism from G to the symmetric group S of X. For more information on this topic see the article on group action. Group representation 115 Representations in other categories Every group G can be viewed as a category with a single object; morphisms in this category are just the elements of G. Given an arbitrary category C, a representation of G in C is a functor from G to C. Such a functor selects an object X in C and a group homomorphism from G to Aut(X), the automorphism group of X. In the case where C is Vect_ the category of vector spaces over a field K, this definition is equivalent to a linear representation. Likewise, a set-theoretic representation is just a representation of G in the category of sets. When C is Ab, the category of abelian groups, the objects obtained are called G-modules. For another example consider the category of topological spaces, Top. Representations in Top are homomorphisms from G to the homeomorphism group of a topological space X. Two types of representations closely related to linear representations are: • projective representations: in the category of projective spaces. These can be described as "linear representations up to scalar transformations". • affine representations: in the category of affine spaces. For example, the Euclidean group acts affinely upon Euclidean space. References • Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics, 129, New York: Springer- Verlag, MR1 153249, ISBN 978-0-387-97527-6, ISBN 978-0-387-97495-8. Introduction to representation theory with emphasis on Lie groups. • Yurii I. Lyubich. Introduction to the Theory of Banach Representations of Groups. Translated from the 1985 Russian-language edition (Kharkov, Ukraine). Birkhauser Verlag. 1988. Unitary representation In mathematics, a unitary representation of a group G is a linear representation jt of G on a complex Hilbert space V such that n(g) is a unitary operator for every g G G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group and the representations are strongly continuous. The theory has been widely applied in quantum mechanics since the 1920s, particularly influenced by Hermann Weyl's 1928 book Gruppentheorie und Quantenmechanik. One of the pioneers in constructing a general theory of unitary representations, for any group G rather than just for particular groups useful in applications, was George Mackey. Context in harmonic analysis The theory of unitary representations of groups is closely connected with harmonic analysis. In the case of an abelian group G, a fairly complete picture of the representation theory of G is given by Pontryagin duality. In general, the unitary equivalence classes of irreducible unitary representations of G make up its unitary dual. This set can be identified with the spectrum of the C*-algebra associated to G by the group C*-algebra construction. This is a topological space. 2 The general form of the Plancherel theorem tries to describe the regular representation of G on L (G) by means of a measure on the unitary dual. For G abelian this is given by the Pontryagin duality theory. For G compact, this is done by the Peter-Weyl theorem; in that case the unitary dual is a discrete space, and the measure attaches an atom to each point of mass equal to its degree. Unitary representation 116 Formal definitions Let G be a topological group. A strongly continuous unitary representation of G on a Hilbert space H is a group homomorphism from G into the unitary group of H, 7T : G -»■ U(#) such that g — > Jt(g) ^ is a norm continuous function for every '%€ H. Note that if G is a Lie group, the Hilbert space also admits underlying smooth and analytic structures. A vector "E, in H is said to be smooth or analytic if the map g — > n(g) % is smooth or analytic (in the norm or weak topologies on H). Smooth vectors are dense in H by a classical argument of Lars Garding, since convolution by smooth functions of compact support yields smooth vectors. Analytic vectors are dense by a classical argument of Edward Nelson, amplified by Roe Goodman, since vectors in the image of a heat operator e , corresponding to an elliptic differential operator D in the universal enveloping algebra of G, are analytic. Not only do smooth or analytic vectors form dense subspaces; they also form common cores for the unbounded skew-adjoint operators corresponding to the T21 elements of the Lie algebra, in the sense of spectral theory. Complete reducibility A unitary representation is completely reducible, in the sense that for any closed invariant subspace, the orthogonal complement is again a closed invariant subspace. This is at the level of an observation, but is a fundamental property. For example, it implies that finite dimensional unitary representations are always a direct sum of irreducible representations, in the algebraic sense. Since unitary representations are much easier to handle than the general case, it is natural to consider unitarizable representations, those that become unitary on the introduction of a suitable complex Hilbert space structure. This works very well for finite groups, and more generally for compact groups, by an averaging argument applied to an arbitrary hermitian structure. For example, a natural proof of Maschke's theorem is by this route. Unitarizability and the unitary dual question In general, for non-compact groups, it is a more serious question which representations are unitarizable. One of the important unsolved problems in mathematics is the description of the unitary dual, the effective classification of irreducible unitary representations of all real reductive Lie groups. All irreducible unitary representations are admissible (or rather their Harish-Chandra modules are), and the admissible representations are given by the Langlands classification, and it is easy to tell which of them have a non-trivial invariant sesquilinear form. The problem is that it is in general hard to tell when this form is positive definite. For many reductive Lie groups this has been solved; see representation theory of SL2(R) and representation theory of the Lorentz group for examples. Notes [1] Warner (1972) [2] Reed and Simon (1975) References • Reed, Michael; Simon, Barry (1975), Methods of Modern Mathematical Physics, Vol. 2: Fourier Analysis, Self-Adjointness, Academic Press, ISBN 0125850026 • Warner, Garth (1972), Harmonic Analysis on Semi-simple Lie Groups I, Springer- Verlag, ISBN 0387054685 Representation theory of the Lorentz group 117 Representation theory of the Lorentz group The Lorentz group of theoretical physics has a variety of representations, corresponding to particles with integer and half-integer spins in quantum field theory. These representations are normally constructed out of spinors. The group may also be represented in terms of a set of functions defined on the Riemann sphere, these are the Riemann P-functions, which are expressible as hypergeometric functions. An important special case is the subgroup SO(3), where these reduce to the spherical harmonics, and find practical application in the theory of atomic spectra. The Lorentz group has no unitary representation of finite dimension, except for the trivial representation (where every group element is represented by 1). Finding representations According to general representation theory of Lie groups, one first looks for the representations of the complexification of the Lie algebra of the Lorentz group. A convenient basis for the Lie algebra of the Lorentz group is given by the three generators of rotations J =E y L.. and the three generators of boosts K^L. where i, j, and k run over the three spatial coordinates and e is the Levi-Civita symbol for a three dimensional spatial slice of Minkowski space. Note that the three generators of rotations transform like components of a pseudovector J and the three generators of boosts transform like components of a vector K under the adjoint action of the spatial rotation subgroup. This motivates the following construction: first complexify, and then change basis to the components of A = (J + i K)/2 and B = (J — i K)/2. In this basis, one checks that the components of A and B satisfy separately the commutation relations of the Lie algebra si and moreover that they commute with each other. In other words, one has the isomorphism so(3, 1) <8> C = sl 2 (C) © sl 2 (C). The utility of this isomorphism comes from the fact that si is the complexification of the rotation algebra, and so its irreducible representations correspond to the well-known representations of the spatial rotation group; for each j in ViL, one has the (2/+l)-dimensional spin-j representation spanned by the spherical harmonics with j as the highest weight. Thus the finite dimensional irreducible representations of the Lorentz group are simply given by an ordered pair of half-integers (m,n) which fix representations of the subalgebra spanned by the components of A and B respectively. Properties of the (m,n) irrep Since the angular momentum operator is given by J = A + B, the highest weight of the rotation subrepresentation will be m+n. So for example, the (1/2,1/2) representation has spin 1. The (m,«) representation is (2ra+ 1 )(2«+ 1 )-dimensional . Common representations • (0,0) the Lorentz scalar representation. This representation is carried by relativistic scalar field theories. • (1/2,0) is the left-handed Weyl spinor and (0,1/2) is the right-handed Weyl spinor representation. • (1/2,0) ® (0,1/2) is the bispinor representation (see also Dirac spinor). • (1/2,1/2) is the four- vector representation. The electromagnetic vector potential lives in this rep. It is a 1-form field. • (1,0) is the self-dual 2-form field representation and (0,1) is the anti-self-dual 2-form field representation. • (1,0) © (0,1) is the representation of a parity invariant 2-form field. The electromagnetic field tensor transforms under this representation. Representation theory of the Lorentz group 118 • (1,1/2) © (1/2,1) is the Rarita-Schwinger field representation. • (1,1) is the spin-2 representation of the traceless metric tensor. Full Lorentz group The (m,«) representation is irreducible under the restricted Lorentz group (the identity component of the Lorentz group). When one considers the full Lorentz group, which is generated by the restricted Lorentz group along with time and parity reversal, not only is this not an irreducible representation, it is not a representation at all, unless m=n. The reason is that this representation is formed in terms of the sum of a vector and a pseudovector, and a parity reversal changes the sign of one, but not the other. The upshot is that a vector in the (m,«) representation is carried into the («,m) representation by a parity reversal. Thus {m,n)®{n,m) gives an irrep of the full Lorentz group. When constructing theories such as QED which is invariant under parity reversal, Dirac spinors may be used, while theories that do not, such as the electroweak force, must be formulated in terms of Weyl spinors. Infinite dimensional unitary representations History The Lorentz group SO(3,l) and its double cover SL(2,C) also have infinite dimensional unitary representations, first studied independently by Bargmann (1947), Gelfand & Naimark (1947) and Harish-Chandra (1947) (at the instigation of Paul Dirac). The Plancherel formula for these groups was first obtained by Gelfand and Naimark through involved calculations. The treatment was subsequently considerably simplified by Harish-Chandra (1951) and Gelfand & Graev (1953), based on an analogue for SL(2,C) of the integration formula of Hermann Weyl for compact Lie groups. Elementary accounts of this approach can be found in Ruhl (1970) and Knapp (2001). The theory of spherical functions for the Lorentz group, required for harmonic analysis on the 3-dimensional hyperboloid in Minkowski space, or equivalently 3-dimensional hyperbolic space, is considerably easier than the general theory. It only involves representations from the spherical principal series and can be treated directly, because in radial coordinates the Laplacian on the hyperboloid is equivalent to the Laplacian on R. This theory is discussed in Takahashi (1963), Helgason (1968), Helgason (2000) and the posthumous text of Jorgenson & Lang (2008). Principal series The principal series, or unitary principal series, are the unitary representations induced from the one-dimensional representations of the lower triangular subgroup B of G = SL(2,C). Since the one-dimensional representations of B correspor the form correspond to the representations of the diagonal matrices, with non-zero complex entries z and z , and thus have for k an integer and v real. The representations are irreducible; the only repetitions occur when k is replaced by — k. 2 2 By definition the representations are realised on L sections of line bundles on G I B = S , the Riemann sphere. When k = 0, these representations constitute the so-called spherical principal series. The restriction of a principal series to the maximal compact subgroup K = SU(2) of G can also be realised as an induced representation of K using the identification G I B = K I T, where T = B f] K is the maximal torus in K consisting of diagonal matrices with Lzl=l. It is the representation induced from the 1 — dimensional representation z T, and is independent of v. By Frobenius reciprocity, on K they decompose as a direct sum of the irreducible representations of K with dimensions \k\ + 2m +1 with m a non-negative integer. Using the identification between the Riemann sphere minus a point and C, the principal series can be defined 2 directly on L (C) by the formula Representation theory of the Lorentz group 119 (a b\ .. x . , 7l _ 2 _i„ ( cz + d \~ faz + b Irreducibility can be checked in a variety of ways: • The representation is already irreducible on B. This can be seen directly, but is also a special case of general results on ireducibility of induced representations due to Francois Bruhat and George Mackey, relying on the Bruhat decomposition G = B U B s B where s is the Weyl group element -l M i] 1 • The action of the Lie algebra gof G can be computed on the algebraic direct sum of the irreducible subspaces of K can be computed explicitly and the it can be verified directly that the lowest dimensional subspace generates this direct sum as a Q -module. Complementary series 2 The for < t < 2, the complementary series is defined on L functions /on C for the inner product (/,«) = // f(z)g(w) dzdw \z — wr with the action given by -1 a b\ n , x , .i-2-i * I az m. A m = \cz+d\- 2 - i f c dj \cz + d; The complementary series are irreducible and inequivalent. As a representation of K, each is isomorphic to the Hilbert space direct sum of all the odd dimensional irreducible representations of K = SU(2). Irreducibility can be proved by analysing the action of gon the algebraic sum of these subspaces or directly without using the Lie algebra. Plancherel theorem The only irreducible unitary representations of SL(2,C) are the principal series, the complementary series and the trivial representation. Since — / acts ( —1) on the principal series and trivially on the remainder, these will give all the irreducible unitary representations of the Lorentz group, provided k is taken to be even. 2 To decompose the left regular representation of G on L (G), only the principal series are required. This immediately 2 yields the decomposition on the subrepresentations L (G/ ±1), the left regular representation of the Lorentz group, 2 and L (G / K), the regular representation on 3-dimensional hyperbolic space. (The former only involves principal series representations with k even and the latter only those with k = 0.) 2 The left and right regular representation X and p are defined on L (G) by %)/(*) = /Or 1 *), p(g)m = f(xg). if/is an element of C (G), the oj T*,fc(/) = / f(9)ir{g)dg JG Now iff is an element of C (G), the operator jt (f) defined by is Hilbert-Schmidt. We define a Hilbert space H by H = HS{L\C)) <g> L 2 {R, c k {u 2 + fc 2 ) 1/2 <H fe>0 where c = 1/47T 3 / 2 , c k = 1/(2tt) 3 / 2 (A; ^ 0) and HS(L (C)) denotes the Hilbert space of Hilbert-Schmidt operators on L (C). Then the map U defined on C c (G) by U{f){ V ,k)=ir Vik {f) Representation theory of the Lorentz group 120 2 extends to a unitary of L (G) onto H. The map U satisfies If/,,/, are in C (G) then /oo TrK t (/i)v(/2)*)(^ + ^ 2 )^ Thus if /=/. * / * denotes the convolution of/ and/ *, where Kfg) = /^(.ff -1 )' t ' len /oo TrK fc (/))( y 2 + fc 2 )^. The last two displayed formulas are usually referred to as the Plancherel formula and the Fourier inversion formula respectively. The Plancherel formula extends to all/ in L AG). By a theorem of Jacques Dixmier and Paul Malliavin, every function /in C°°(G) is a finite sum of convolutions of similar functions, the inversion formula T71 holds for such/. It can be extended to much wider classes of functions satisfying mild differentiability conditions. See also • Poincare group • Wigner's classification Notes [1] Knapp 2001, Chapter II [2] Harish-Chandra 1947 [3] Taylor 1986 [4] Gelfand & Naimark 1947 [5] Takahashi 1963, p. 343 [6] Note that for a Hilbert space H, HS(H) may be identified canonically with the Hilbert space tensor product of H and its conjugate space. [7] Knapp 2001 References • Bargmann, V. (1947), "Irreducible unitary representations of the Lorenz group" (http://jstor.org/stable/ 1969129), Ann. Of Math. 48 (3): 568-640, doi:10.2307/1969129 (the representation theory of SO(2,l) and SL(2,R); the second part on SO(3,l) and SL(2,C), described in the introduction, was never published). • Dixmier, J.; Malliavin, P. (1978), "Factorisations de fonctions et de vecteurs indefiniment differentiables", Bull. Sc. Math. 102: 305-330 • Gelfand, I. M.; Naimark, M. A. (1947), "Unitary representations of the Lorentz group", Izvestiya Akad. Nauk SSSR. Ser. Mat. 11: 411-504 • Gelfand, I. M.; M. I. Graev (1953), Doklady Akademii Nauk SSSR 92: 221-224 • Harish-Chandra (1947), "Infinite irreducible representations of the Lorentz group", Proc. Roy. Soc. London. Ser. A. 189: 372-401, doi: 10. 1098/rspa. 1947.0047 • Harish-Chandra (1951), "Plancherel formula for complex semi-simple Lie groups", Proc. Nat. Acad. Sci. U. S. A. 37: 813-818, doi:10.1073/pnas.37. 12.813 • Helgason, S. (1968), Lie groups and symmetric spaces, Battelle Rencontres, Benjamin, pp. 1—71 (a general introduction for physicists) • Helgason, S. (2000), Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions (corrected reprint of the 1984 original), Mathematical Surveys and Monographs, 83, American Mathematical Society, ISBN 0-8218-2673-5 Representation theory of the Lorentz group 121 • Jorgenson, J.; Lang, S. (2008), The heat kernel and theta inversion on SL(2,C), Springer Monographs in Mathematics, Springer, ISBN 978-0-387-38031-5 • Knapp, A. (2001), Representation theory of semisimple groups. An overview based on examples., Princeton Landmarks in Mathematics, Princeton University Press, ISBN 0-691-09089-0 (elementary treatment for SL(2,C)) • Naimark, M.A. (1964), Linear representations of the Lorentz group (translated from the Russian original by Ann Swinfen and O. J. Marstrand), Macmillan • Paerl, E.R. (1969) Representations of the Lorentz group and projective geometry, Mathematical Centre Tract #25, Amsterdam. • Ruhl, W. (1970), The Lorentz group and harmonic analysis, Benjamin (a detailed account for physicists) • Takahashi, R. (1963), "Sur les representations unitaires des groupes de Lorentz generalises", Bull. Soc. Math. France 91: 289-433 • Taylor, M.E. (1986), Noncommutative harmonic analyis, Mathematical Surveys and Monographs, 22, American Mathematical Society, ISBN 0-8218-1523-7, Chapter 9, SL(2,C) and more general Lorentz groups Stone-von Neumann theorem In mathematics and in theoretical physics, the Stone— von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. The name is for Marshall Stone and von Neumann (1931). Trying to represent the commutation relations In quantum mechanics, physical observables are represented mathematically by linear operators on Hilbert spaces. For a single particle moving on the real line R, there are two important observables: position and momentum. In the quantum-mechanical description of such a particle, the position operator Q and momentum operator P are respectively given by [Qlp](x) — Xlp(x) [P1i\{x) = V(z) on the domain V of infinitely differentiable functions of compact support on R. We assume /j, is a fixed non-zero real number — in quantum theory /j, is (up to a factor of 2jt) Planck's constant, which is not dimensionless; it takes a small numerical value in terms of units in the macroscopic world. The operators P, Q satisfy the commutation relation QP-PQ=--1. i Already in his classic volume, Hermann Weyl observed that this commutation law was impossible for linear operators P, Q acting on finite dimensional spaces (as is clear by applying the trace of a matrix), unless /j, vanishes. A little analysis shows that in fact any two self-adjoint operators satisfying the above commutation relation cannot be both bounded. Uniqueness of representation One would like to classify representations of the canonical commutation relation by two self-adjoint operators acting on separable Hilbert spaces, up to unitary equivalence. By Stone's theorem, there is a one-to-one correspondence between self-adjoint operators and (strongly continuous) one parameter unitary groups. Let Q and P be two self-adjoint operators satisfying the canonical commutation relation, and e 1 and e 1,s be the corresponding unitary groups given by functional calculus. A formal computation with power series shows that Stone— von Neumann theorem 122 e itQ e isP _ e ist e isP e itQ _ q Conversely, given two one parameter unitary groups U(t) and V(s) satisfying the relation U(t)V(s) = e ist V(s)U(t) Vs, t, (*) formally differentiating at shows that the two infinitesmal generators satisfy the canonical commutation relation. These formal calculations can be made rigorous. Therefore there is a one-to-one correspondence between representations of the canonical commutation relation and two one parameter unitary groups U(t) and V(s) satisfying (*). This formulation of the canonical commutation relations for one parameter unitary groups is called the Weyl form of the CCR. The problem now thus becomes classifying two jointly irreducible one parameter unitary groups U(t) and V(s) satisfying the Weyl relation on separable Hilbert spaces. The answer is the content of the Stone-von Neumann theorem: all such pairs of one parameter unitary groups are unitarily equivalent. In other words, for any two such U(t) and V(s) acting jointly irreducibly on a Hilbert space H, there is a unitary operator W :L 2 (R) -># so that W*U(t)W = e isQ and W*V(t)W = e isP , where P and Q are the position and momentum operators from above. Historically this result was significant because it was a key step in proving that Heisenberg's matrix mechanics which presents quantum mechanical observables and dynamics in terms of infinite matrices, is unitarily equivalent to Schrodinger's wave mechanical formulation (see Schrodinger picture). Representation theory formulation In terms of representation theory, the Stone— von Neumann theorem classifies certain unitary representations of the Heisenberg group. This is discussed in more detail in the Heisenberg group section, below. Informally stated, with certain technical assumptions, every representation of the Heisenberg group i^^+iis equivalent to the position operators and momentum operators on R . Alternatively, that they are all equivalent to the Weyl algebra (or CCR algebra) on a symplectic space of dimension 2n. More formally, there is a unique (up to scale) non-trivial central strongly continuous unitary representation. This was later generalized by Mackey theory — and was the motivation for the introduction of the Heisenberg group in quantum physics. In detail: 2n • The continuous Heisenberg group is a central extension of the abelian Lie group R by a copy of R, 2n • the corresponding Heisenberg algebra is a central extension of the abelian Lie algebra R (with trivial bracket) by a copy of R, 2n • the discrete Heisenberg group is a central extension of the free abelian group Z by a copy of Z, and 2n • the discrete Heisenberg group module p is a central extension of the free abelian p-group {XI pL) by a copy of TJpZ. These are thus all semidirect product, and hence relatively easily understood. In all cases, if one has a representation JJ — j. A where the center maps to zero, then one simply has a representation of the corresponding abelian group or algebra, which is Fourier theory. If the center does not map to zero, one has a more interesting theory, particularly if one restricts oneself to central representations. Concretely, by a central representation one means a representation such that the center of the Heisenberg group maps into the center of the algebra: for example, if one is studying matrix representations or representations by operators on a Hilbert space, then the center of the matrix algebra or the operator algebra is the scalar matrices. Thus the Stone— von Neumann theorem 123 representation of the center of the Heisenberg group is determined by a scale value, called the quantization value (in physics terms, Planck's constant), and if this goes to zero, one gets a representation of the abelian group (in physics terms, this is the classical limit). More formally, the group algebra of the Heisenberg group JiffHlhas center .K"[R,1 , so rather than simply thinking of the group algebra as an algebra over the field of scalars K, one may think of it as an algebra over the commutative algebra .KTR.]. As the center of a matrix algebra or operator algebra is the scalar matrices, a iiT [R.1 -structure on the matrix algebra is a choice of scalar matrix — a choice of scale. Given such a choice of scale, a central representation of the Heisenberg group is a map of iiT [R.1 -algebras iiffi/] — > A, which is the formal way of saying that it sends the center to a chosen scale. Then the Stone— von Neumann theorem is that, given a quantization value, every strongly continuous unitary representation is unitarily equivalent to the standard representation as position and momentum. Reformulation via Fourier transform A Let G be a locally compact abelian group and G be the Pontryagin dual of G. The Fourier-Plancherel transform defined by / ^ f(l) = / l(t)f(tW(t) JG extends to a C*-isomorphism from the group C*-algebra C*(G) of G and C AG ), i.e. the spectrum of C*(G) is A precisely G . When G is the real line R, this is Stone's theorem characterizing one parameter unitary groups. The theorem of Stone-von Neumann can also be restated using similar language. The group G acts on the C*-algebra C AG) by right translation g: for s in G and/in C AG), (s.f)(t) = f(t + s). A Under the isomorphism given above, this action becomes the natural action of G on C*(G ): M(t) = tW/(t). So a covariant representation corresponding to the C*-crossed product <T{6) XpG A is a unitary representation U(s) of G and V(y) of G such that tf(a)V( 7 )l/*00=700V(7). It is a general fact that covariant representations are in one-to-one correspondence with *-representation of the corresponding crossed product. On the other hand, all irreducible representations of C (G) x p G 2 2 are unitarily equivalent to the K(L( G)), the compact operators on L( G)). Therefore all pairs {U(s), V(y)} are unitarily equivalent. Specializing to the case where G = R yields the Stone-von Neumann theorem. The Heisenberg group The commutation relations for P, Q look very similar to the commutation relations that define the Lie algebra of general Heisenberg group H for n a positive integer. This is the Lie group of (n+2) x (n+2) square matrices of the form lac M(a, b,c)= 1„ b [o 1 In fact, using the Heisenberg group, we can formulate a far-reaching generalization of the Stone von Neumann theorem. Note that the center of H consists of matrices M(0, 0, c). Stone— von Neumann theorem 124 Theorem. For each non-zero real number h there is an irreducible representation U acting on the Hilbert space L 2 (R") by [U h (M(a, b, c))]1>{x) = e^'+^ix + ha). All these representations are unitarily inequivalent and any irreducible representation which is not trivial on the center of H is unitarily equivalent to exactly one of these. Note that U is a unitary operator because it is the composition of two operators which are easily seen to be unitary: the translation to the left by h a and multiplication by a function of absolute value 1 . To show U is multiplicative is a straightforward calculation. The hard part of the theorem is showing the uniqueness which is beyond the scope of the article. However, below we sketch a proof of the corresponding Stone— von Neumann theorem for certain finite Heisenberg groups. In particular, irreducible representations n, it' of the Heisenberg group H which are non-trivial on the center of H are unitarily equivalent if and only if n(z) = JT'(z) for any z in the center of H . One representation of the Heisenberg group that is important in number theory and the theory of modular forms is the theta representation, so named because the Jacobi theta function is invariant under the action of the discrete subgroup of the Heisenberg group. Relation to the Fourier transform For any non-zero h, the mapping a h : M(a, b, c) -> M(-/i -1 &, ha, c - ab) is an automorphism of H which is the identity on the center of H .In particular, the representations U and U a are unitarily equivalent. This means that there is a unitary operator WonL (R n ) such that for any g in H , WU h (g)W* = U h a(g). Moreover, by irreducibility of the representations U , it follows that up to a scalar, such an operator W is unique (cf. Schur's lemma). 2 n Theorem. The operator Wis, up to a scalar multiple, the Fourier transform on L (R ). nil This means that (ignoring the factor of (2 it) in the definition of the Fourier transform) I e- ix - p e i( - b - x+hc ^(x + ha) dx = e ^a- P +h(c-b-a)) j e -™/(*-b)^( y ) dy The previous theorem can actually be used to prove the unitary nature of the Fourier transform, also known as the Plancherel theorem. Moreover, note that (a h ) 2 M(a, 6, c) = M(-a, -b, c). Theorem. The operator W such that W 1 U h W* = U h a 2 {g) is the reflection operator [W 1 il>]{x)=,l } {-x). From this fact the Fourier inversion formula easily follows. Stone— von Neumann theorem 125 Representations of finite Heisenberg groups The Heisenberg group H (K) is defined for any commutative ring K. In this section let us specialize to the field K = XI p Z for p a prime. This field has the property that there is an imbedding m of K as an additive group into the circle group T. Note that H (K) is finite with cardinality IKI . For finite Heisenberg group H (K) one can give a simple proof of the Stone— von Neumann theorem using simple properties of character functions of representations. These properties follow from the orthogonality relations for characters of representations of finite groups. 2 n For any non-zero h in K define the representation U on the finite-dimensional inner product space / (K ) by It follows that [U h M(a, b, c)ip](x) = u(b ■ x + hc)i>{x + ha). em. For a fixed non-zero h, the character function x of U is given by: „,, L „ (\K\ n u>(hc) if a = 6 = x(M(a,6 )C )) = |' / otherw . se _ hat g€H„(K) By the orthogonality relations for characters of representations of finite groups this fact implies the corresponding Stone— von Neumann theorem for Heisenberg groups H (Z/p Z), particularly: • Irreducibility of U • Pairwise inequivalence of all the representations U . Generalizations The Stone— von Neumann theorem admits numerous generalizations. Much of the early work of George Mackey was directed at obtaining a formulation of the theory of induced representations developed originally by Frobenius for finite groups to the context of unitary representations of locally compact topological groups. References • Kirillov, A. A. (1976), Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften, 220, Berlin, New York: Springer-Verlag, MR0407202, ISBN 978-0-387-07476-4 • G. W. Mackey, The Theory of Unitary Group Representations, The University of Chicago Press, 1976 • von Neumann, John (1931), "Die Eindeutigkeit der Schrodingerschen Operatoren" , Mathematische Annalen (Springer Berlin / Heidelberg) 104: 570-578, doi:10.1007/BF01457956, ISSN 0025-5831 • H. Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications, 1950 References [1] http://dx.doi.org/10.1007/BF01457956 Peter— Weyl theorem 126 Peter— Weyl theorem In mathematics, the Peter— Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, in the setting of a compact topological group G (Peter & Weyl 1927). The theorem is a collection of results generalizing the significant facts about the decomposition of the regular representation of any finite group, as discovered by F. G. Frobenius and Issai Schur. The theorem has three parts. The first part states that the matrix coefficients of G are dense in the space C(G) of 2 continuous complex-valued functions on G, and thus also in the space L (G) of square-integrable functions. The second part asserts the complete reducibility of unitary representations of G. The third part then asserts that the 2 regular representation of G on L (G) decomposes as the direct sum of all irreducible unitary representations. 2 Moreover, the matrix coefficients of the irreducible unitary representations form an orthonormal basis of L (G). Matrix coefficients A matrix coefficient of the group G is a complex-valued function (j) ; on G given as the composition ip = L O 7T where n : G — » GL(V) is a finite-dimensional (continuous) group representation of G, and L is a linear functional on the vector space of endomorphisms of V (e.g. trace), which contains GL(V) as an open subset. Matrix coefficients are continuous, since representations are by definition continuous, and linear functionals on finite-dimensional spaces are also continuous. The first part of the Peter— Weyl theorem asserts (Bump 2004, §4.1; Knapp 1986, Theorem 1.12): • The set of matrix coefficients of G is dense in the space of continuous complex functions C(G) on G, equipped with the uniform norm. This first result resembles the Stone-Weierstrass theorem in that it indicates the density of a set of functions in the space of all continuous functions, subject only to an algebraic characterization. In fact, if G is a matrix group, then the result follows easily from the Stone-Weierstrass theorem (Knapp 1986, p. 17). Conversely, it is a consequence of the subsequent conclusions of the theorem that any compact Lie group is isomorphic to a matrix group (Knapp 1986, Theorem 1.15). 2 A corollary of this result is that the matrix coefficients of G are dense in L (G). Decomposition of a unitary representation The second part of the theorem gives the existence of a decomposition of a unitary representation of G into finite-dimensional representations. Now, intuitively groups were conceived as rotations on geometric objects, so it is only natural to study representations which essentially arise from continuous actions on Hilbert spaces. (For those who were first introduced to dual groups consisting of characters which are the continuous homomorphisms into the circle group {zeC:|z|=l} , this approach is similar except that the circle group is (ultimately) generalised to the group of unitary operators on a given Hilbert space.) Let G be a topological group and H a complex Hilbert space. Given a continuous action *:GxH^H , it gives rise to a map p*-.G—> H H defined in the obvious way: p»(v)=gv . This map is clearly an homomorphism from G into GL(H), the homeomorphic automorphisms of H. And given such a map, we can uniquely recover the action in the obvious way. Thus we define the representations of G on an Hilbert space H to be those group homomorphisms, p, which arise from continuous actions of G on H. We say that a representation p is unitary if p(g) is a unitary operator for all g € G; i.e., {gv,g-w)={v,-m) for all v, w G H. (I.e. it is unitary if p:G—>U(H). Notice how this generalises the special Peter— Weyl theorem 127 case of the one-dimensional Hilbert space, where {/(C 1 ) is just the circle group.) Given these definitions, we can state the Peter— Weyl theorem. The second part of this theorem asserts (Knapp 1986, Theorem 1.14): • Let p be a unitary representation of a compact group G on a complex Hilbert space H. Then H splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of G. Decomposition of square-integrable functions To state the third and final part of the theorem, there is a natural Hilbert space over G consisting of square-integrable 2 functions, L (G); this makes sense because Haar measure exists on G. Calling this Hilbert space H, the group G has a unitary representation p on H by acting on the left, via p(h)f(g)=f(h- 1 g). The final statement of the Peter— Weyl theorem (Knapp 1986, Theorem 1.14) gives an explicit orthonormal basis of 2 2 L (G). Roughly it asserts that the matrix coefficients for G, suitably renormalized, are an orthonormal basis of L (G). 2 In particular, L (G) decomposes into an orthogonal direct sum of all the irreducible unitary representations, in which the multiplicity of each irreducible representation is equal to its degree (that is, the dimension of the underlying space of the representation). Thus, L\G) = ^ Ti-eE where 2 denotes the set of (isomorphism classes of) irreducible unitary representations of G, and the summation denotes the closure of the direct sum of the total spaces E of the representations it. More precisely, suppose that a representative jt is chosen for each isomorphism class of irreducible unitary representation, and denote the collection of all such ji by 2. Let u ^be the matrix coefficients of jt in an orthonormal basis, in other words u i]\9) = (^(9)ei,e j ). for each g G G Finally, let <r be the degree of the representation jr. The theorem now asserts that the set of functions {VdF)u\] ] | 7T G E, 1 < i,j < d^} 2 is an orthonormal basis of L (G). Consequences Structure of compact topological groups From the theorem, one can deduce a significant general structure theorem. Let G be a compact topological group, 2 which we assume Hausdorff. For any finite-dimensional G-invariant subspace V in L (G), where G acts on the left, we consider the image of G in GL(V). It is closed, since G is compact, and a subgroup of the Lie group GL(V). It follows by a theorem of Elie Cartan that the image of G is a Lie group also. If we now take the limit (in the sense of category theory) over all such spaces V, we get a result about G - because G 2 acts faithfully on L (G). We can say that G is group: it may for example be a profinite group. 2 acts faithfully on L (G). We can say that G is an inverse limit of Lie groups. It may of course not itself be a Lie Peter— Weyl theorem 128 References • Peter, F.; Weyl, H. (1927), "Die Vollstandigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe", Math. Ann. 97: 737-755, doi:10.1007/BF01447892. • Palais, R.S.; Stewart, T.E. (1961), "The cohomology of differentiable transformation groups" ,Amer. J. Math. (The Johns Hopkins University Press) 83 (4): 623-644, doi:10.2307/2372901. • Knapp, Anthony (1986), Representation theory of semisimple groups, Princeton University Press, ISBN 0-691-09089-0. • Bump, Daniel (2004), Lie groups, Springer, ISBN 0-387-21 154-3. 121 • Mostow, G.D. (1961), "Cohomology of topological groups and solvmanifolds" ,Ann. Of Math. (Annals of Mathematics) 73 (1): 20-48, doi: 10.2307/1970281 References [1] http://www.jstor.org/stable/2372901 [2] http://www.jstor.org/stable/1970281 Quantum algebra Quantum algebra is one of the top-level mathematics categories used by the arXiv. Subjects include: • Quantum groups • Skein theories • Operadic algebra • Diagrammatic algebra • Quantum field theory External links • Quantum algebra at arxiv.org References [1] http://arxiv.org/list/math.QA/current Quantum affine algebra 129 Quantum affine algebra In mathematics, a quantum affine algebra (or affine quantum group) is a Hopf algebra that is a ^-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced independently by Drinfeld (1985) and Jimbo (1985) as a special case of their general construction of a quantum group from a Cartan matrix. One of their principal applications has been to the theory of solvable lattice models in quantum statistical mechanics, where the Yang-Baxter equation occurs with a spectral parameter. Combinatorial aspects of the representation theory of quantum affine algebras can be described simply using crystal bases, which correspond to the degenerate case when the deformation parameter q vanishes and the Hamiltonian of the associated lattice model can be explicitly diagonalized. References • Drinfeld, V. G. (1985), "Hopf algebras and the quantum Yang-Baxter equation", Doklady Akademii Nauk SSSR 283 (5): 1060-1064, MR802128, ISSN 0002-3264 • Drinfeld, V. G. (1987), "A new realization of Yangians and of quantum affine algebras", Doklady Akademii Nauk SSSR 296 (1): 13-17, MR914215, ISSN 0002-3264 • Frenkel, Igor B.; Reshetikhin, N. Yu. (1992), "Quantum affine algebras and holonomic difference equations" , Communications in Mathematical Physics 146 (1): 1-60, doi:10.1007/BF02099206, MR1163666, ISSN 0010-3616 • Jimbo, Michio (1985), "A q-difference analogue of U(g) and the Yang-Baxter equation", Letters in Mathematical Physics. 10 (1): 63-69, doi:10.1007/BF00704588, MR797001, ISSN 0377-9017 • Jimbo, Michio; Miwa, Tetsuji (1995), Algebraic analysis of solvable lattice models, CBMS Regional Conference Series in Mathematics, 85, Published for the Conference Board of the Mathematical Sciences, Washington, DC, MR1308712, ISBN 978-0-8218-0320-2 Clifford algebra 130 Clifford algebra In mathematics, Clifford algebras are a type of associative algebra. They can be thought of as one of the possible generalizations of the complex numbers and quaternions. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry and theoretical physics. They are named after the English geometer William Kingdon Clifford. Introduction and basic properties Specifically, a Clifford algebra is a unital associative algebra which contains and is generated by a vector space V equipped with a quadratic form Q. The Clifford algebra C({V,Q) is the "freest" algebra generated by V subject to the condition v 2 = Q(v)l for all v G V. If the characteristic of the ground field K is not 2, then one can rewrite this fundamental identity in the form uv + vu — 2(u, v) for all u, v G V, where <u, v> = Vi(Q(u + v) - Q(u) - Q{v)) is the symmetric bilinear form associated to Q, via the polarization identity. The idea of being the "freest" or "most general" algebra subject to this identity can be formally expressed through the notion of a universal property, as done below. Quadratic forms and Clifford algebras in characteristic 2 form an exceptional case. In particular, if char K = 2 it is not true that a quadratic form determines a symmetric bilinear form, or that every quadratic form admits an orthogonal basis. Many of the statements in this article include the condition that the characteristic is not 2, and are false if this condition is removed. As quantization of exterior algebra Clifford algebras are closely related to exterior algebras. In fact, if Q = then the Clifford algebra C({V,Q) is just the exterior algebra A(V). For nonzero Q there exists a canonical linear isomorphism between A(V) and C({V,Q) whenever the ground field K does not have characteristic two. That is, they are naturally isomorphic as vector spaces, but with different multiplications (in the case of characteristic two, they are still isomorphic as vector spaces, just not naturally). Clifford multiplication is strictly richer than the exterior product since it makes use of the extra information provided by Q. More precisely, Clifford algebras may be thought of as quantizations (cf. quantization (physics), Quantum group) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra. Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras. Universal property and construction Let V be a vector space over a field K, and let Q : V — > K be a quadratic form on V. In most cases of interest the field K is either R, C or a finite field. A Clifford algebra C({V,Q) is a unital associative algebra over K together with a linear map i : V — > C({V,Q) satisfying i(v) = Q(v)l for all v £ V, defined by the following universal property: Given any associative algebra A over K and any linear map j : V — > A such that j(v) 2 =g(v)l for allv€ V Clifford algebra 131 (where 1 denotes the multiplicative identity of A), there is a unique algebra homomorphism / : C€{V,Q) — > A such that the following diagram commutes (i.e. such that/o i =j): i i i :./ ■ * A Working with a symmetric bilinear form <■,■> instead of Q (in characteristic not 2), the requirement on j is j(v)j(w) +j(w)j(v) = 2<v, w> for all v, w G V. A Clifford algebra as described above always exists and can be constructed as follows: start with the most general algebra that contains V, namely the tensor algebra T(V), and then enforce the fundamental identity by taking a suitable quotient. In our case we want to take the two-sided ideal /„in T(V) generated by all elements of the form v ® v - Q(v)Uot all v £ V and define C({V,Q) as the quotient C((V,Q) = T(V)/I Q It is then straightforward to show that C({V,Q) contains V and satisfies the above universal property, so that CI is unique up to a unique isomorphism; thus one speaks of "the" Clifford algebra C({V, Q). It also follows from this construction that i is injective. One usually drops the i and considers V as a linear subspace of C({V,Q). The universal characterization of the Clifford algebra shows that the construction of C({V,Q) is functorial in nature. Namely, CI can be considered as a functor from the category of vector spaces with quadratic forms (whose morphisms are linear maps preserving the quadratic form) to the category of associative algebras. The universal property guarantees that linear maps between vector spaces (preserving the quadratic form) extend uniquely to algebra homomorphisms between the associated Clifford algebras. Basis and dimension If the dimension of Vis n and {e ,,..., e } is a basis of V, then the set 1 n {e^e^ ■ ■ ■ Ci k | 1 < ii < i2 < ■ ■ ■ < ik < n and < k < n} is a basis for C({V,Q). The empty product (k = 0) is defined as the multiplicative identity element. For each value of k there are n choose k basis elements, so the total dimension of the Clifford algebra is fc=0 ^ ^ Since V comes equipped with a quadratic form, there is a set of privileged bases for V: the orthogonal ones. An orthogonal basis is one such that (e i? ej) =0 i^j. where <■,■> is the symmetric bilinear form associated to Q. The fundamental Clifford identity implies that for an orthogonal basis ^ 3 3 i / <* ' This makes manipulation of orthogonal basis vectors quite simple. Given a product &i^G-i 2 ' ■ -e^of distinct orthogonal basis vectors, one can put them into standard order by including an overall sign corresponding to the number of flips needed to correctly order them (i.e. the signature of the ordering permutation). If the characteristic is not 2 then an orthogonal basis for V exists, and one can easily extend the quadratic form on V to a quadratic form on all of C({V,Q) by requiring that distinct elements e^e^ ■ ■ ■ e^ are orthogonal to one another whenever the {e.}'s are orthogonal. Additionally, one sets Clifford algebra 132 Qie^e^ ■ ■ -e ik ) = Q{e il )Q{e i2 ) ■ ■ -Q{e ik ). 2 The quadratic form on a scalar is just Q(\) = X . Thus, orthogonal bases for V extend to orthogonal bases for C{(V,Q). The quadratic form defined in this way is actually independent of the orthogonal basis chosen (a basis-independent formulation will be given later). Examples: real and complex Clifford algebras The most important Clifford algebras are those over real and complex vector spaces equipped with nondegenerate quadratic forms. The geometric interpretation of nondegenerate Clifford algebras is known as geometric algebra. Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form: Q(v) = v 2 + --- + v 2 p -v 2 p+1 v 2 p+g where n = p + q is the dimension of the vector space. The pair of integers (p, q) is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted R . The Clifford algebra on R p ' q is denoted CI (R). The symbol CI (R) means either CI „(R) or C€ n (R) depending on whether the author prefers p,q n nfl 0,n positive definite or negative definite spaces. A standard orthonormal basis {e.} for R p ' q consists of n = p + q mutually orthogonal vectors, p of which have norm +1 and q of which have norm -1. The algebra CI (R) will therefore have p vectors which square to +1 and q vectors which square to -1. Note that C£ (R) is naturally isomorphic to R since there are no nonzero vectors. CI (R) is a two-dimensional algebra generated by a single vector e which squares to -1, and therefore is isomorphic to C, the field of complex numbers. The algebra CI n9 (R) is a four-dimensional algebra spanned by {1, e , e , e e }. The latter three elements square to -1 and all anticommute, and so the algebra is isomorphic to the quaternions H. The next algebra in the sequence is CI (R) is an 8 -dimensional algebra isomorphic to the direct sum H © H called split-biquaternions. One can also study Clifford algebras on complex vector spaces. Every nondegenerate quadratic form on a complex vector space is equivalent to the standard diagonal form Q[z) = zl + Z 2 2 + --- + z 2 n where n = dim V, so there is essentially only one Clifford algebra in each dimension. We will denote the Clifford algebra on C with the standard quadratic form by CI (C). One can show that the algebra CI (C) may be obtained as the complexification of the algebra CI (R) where n= p + q: c£ n (c) = ce P!q (R) ®c = ce(c p+q , q ® c). Here Q is the real quadratic form of signature (p,q). Note that the complexification does not depend on the signature. The first few cases are not hard to compute. One finds that « (C) = c c^(C) = c e c « 2 (C) = M 2 (C) where Af (C) denotes the algebra of 2x2 matrices over C. It turns out that every one of the algebras CI (R) and CI (C) is isomorphic to a matrix algebra over R, C, or H or p,q n to a direct sum of two such algebras. For a complete classification of these algebras see classification of Clifford algebras. Clifford algebra 133 Properties Relation to the exterior algebra Given a vector space V one can construct the exterior algebra A(V), whose definition is independent of any quadratic form on V. It turns out that if F does not have characteristic 2 then there is a natural isomorphism between A(V) and C((V,Q) considered as vector spaces (and there exists an isomorphism in characteristic two, which may not be natural). This is an algebra isomorphism if and only if Q = 0. One can thus consider the Clifford algebra C({V,Q) as an enrichment (or more precisely, a quantization, cf. the Introduction) of the exterior algebra on V with a multiplication that depends on Q (one can still define the exterior product independent of Q). The easiest way to establish the isomorphism is to choose an orthogonal basis {e.} for V and extend it to an orthogonal basis for C({V,Q) as described above. The map C({V,Q) — > A(V) is determined by ^i\ ^12 " " " ^k i\ 12 ' ' ' ^k ' Note that this only works if the basis {e.} is orthogonal. One can show that this map is independent of the choice of orthogonal basis and so gives a natural isomorphism. If the characteristic of K is 0, one can also establish the isomorphism by antisymmetrizing. Define functions/ : V x x V -^ Cl(V,Q) by = k\ fk{vi,---,Vk) = jj Y, s g n ( cr ) V '{1) ■ ■ ■ v '(k) where the sum is taken over the symmetric group on k elements. Since/ is alternating it induces a unique linear map k ' A (V) — > C£(V,Q). The direct sum of these maps gives a linear map between A(V) and C£(V,Q). This map can be shown to be a linear isomorphism, and it is natural. A more sophisticated way to view the relationship is to construct a filtration on C€{V,Q). Recall that the tensor algebra T(V) has a natural filtration: F C F C F C ... where F contains sums of tensors with rank < k. Projecting this down to the Clifford algebra gives a filtration on C({V,Q). The associated graded algebra Gr F C£(V,Q) = ($F k /F k - 1 k is naturally isomorphic to the exterior algebra A(V). Since the associated graded algebra of a filtered algebra is always isomorphic to the filtered algebra as filtered vector spaces (by choosing complements of F in r + for all k), this provides an isomorphism (although not a natural one) in any characteristic, even two. Grading In the following, assume that the characteristic is not 2. Clifford algebras are Z -graded algebras (also known as superalgebras). Indeed, the linear map on V defined by V i— » — v (reflection through the origin) preserves the quadratic form Q and so by the universal property of Clifford algebras extends to an algebra automorphism a : C((V,Q) -» C£(V,Q). Since a is an involution (i.e. it squares to the identity) one can decompose Cl{V,Q) into positive and negative eigenspaces C£(V, Q) = Ct°(V, Q) © Ct\V, Q) where C(\V,Q) = {x € C({V,Q) I a(x) = (-l)x}. Since a is an automorphism it follows that Ct{V, Q)C£ j (V, Q) = C£ i+j (V, Q) where the superscripts are read modulo 2. This gives C({V,Q) the structure of a Z -graded algebra. The subspace CI (V,Q) forms a subalgebra of C((V,Q), called the even subalgebra. The subspace CI (V,Q) is called the odd part of C({V,Q) (it is not a subalgebra). The Z -grading plays an important role in the analysis and application of Clifford Clifford algebra 134 algebras. The automorphism a is called the main involution or grade involution. Remark. In characteristic not 2 the underlying vector space of C({V,Q) inherits a Z-grading from the canonical isomorphism with the underlying vector space of the exterior algebra A(V). It is important to note, however, that this is a vector space grading only. That is, Clifford multiplication does not respect the Z-grading, only the Z-grading: for instance if Q(v) / 0, then v £ Cl l (V,Q), but v 2 £ C£°(V,Q), not in C£ 2 (V,Q)- Happily, the gradings are related in the natural way: Z = Z/2Z. Further, the Clifford algebra is Z-filtered: Cl-^V, Q) ■ C£- j (V, Q) C C£- i+j (V, Q)- The degree of a Clifford number usually refers to the degree in the Z-grading. Elements which are pure in the Z-grading are simply said to be even or odd. 2 [3] The even subalgebra CI (V,Q) of a Clifford algebra is itself a Clifford algebra . If V is the orthogonal direct sum of a vector a of norm Q(a) and a subspace U, then CI (V,Q) is isomorphic to C({U,-Q{a)Q), where -Q{a)Q is the form Q restricted to U and multiplied by -Q(a). In particular over the reals this implies that C£° pq (R) = C£ Pi ,_i(R) for q > 0, and C£l q (R) = C£ qtP _ 1 (R)fov P >0. In the negative-definite case this gives an inclusion C( n ,(R) C Ct n (R) which extends the sequence ° ° 0,n-l 0, n J RCCCHCH9HC ... Likewise, in the complex case, one can show that the even subalgebra of CI (C) is isomorphic to CI (C). Antiautomorphisms In addition to the automorphism a, there are two antiautomorphisms which play an important role in the analysis of Clifford algebras. Recall that the tensor algebra T(V) comes with an antiautomorphism that reverses the order in all products: Vi ® v 2 ® ■ ■ ■ ® v k i-> v k ® • ■ ■ <g> v 2 ® v 1 . Since the ideal / is invariant under this reversal, this operation descends to an antiautomorphism of C({V,Q) called the transpose or reversal operation, denoted by x . The transpose is an antiautomorphism: (xyY = y t x t ■ The transpose operation makes no use of the Z-grading so we define a second antiautomorphism by composing a and the transpose. We call this operation Clifford conjugation denoted x x = a(x t ) = a{x) 1 . Of the two antiautomorphisms, the transpose is the more fundamental. Note that all of these operations are involutions. One can show that they act as ±1 on elements which are pure in the Z-grading. In fact, all three operations depend only on the degree modulo 4. That is, if x is pure with degree k then a{x) = ±x x t = ±x x = ±x where the signs are given by the following table: [41 k mod 4 1 2 3 a{x) + - + - (-1)* x l + + - - ,yk{k-\)l2 X + - - + t^k+W Clifford algebra 135 The Clifford scalar product When the characteristic is not 2 the quadratic form Q on V can be extended to a quadratic form on all of C({V,Q) as explained earlier (which we also denoted by Q). A basis independent definition is Q(x) = (x'x) where <a> denotes the scalar part of a (the grade part in the Z-grading). One can show that Q{v 1 v 2 ■■■v k ) = QivijQfa) ■ ■ ■ Q(v k ) where the v. are elements of V — this identity is not true for arbitrary elements of C({V,Q). The associated symmetric bilinear form on C({V,Q) is given by {x,y) = {x l y). One can check that this reduces to the original bilinear form when restricted to V. The bilinear form on all of C({V,Q) is nondegenerate if and only if it is nondegenerate on V. It is not hard to verify that the transpose is the adjoint of left/right Clifford multiplication with respect to this inner product. That is, (ax,y) = (x,a t y) ,and (xa,y) = {x.ya 1 ). Structure of Clifford algebras In this section we assume that the vector space V is finite dimensional and that the bilinear form of Q is non-singular. A central simple algebra over K is a matrix algebra over a (finite dimensional) division algebra with center K. For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions. • If V has even dimension then C({V,Q) is a central simple algebra over K. • If V has even dimension then CI (V,Q) is a central simple algebra over a quadratic extension of K or a sum of two isomorphic central simple algebras over K. • If V has odd dimension then C({V,Q) is a central simple algebra over a quadratic extension of K or a sum of two isomorphic central simple algebras over K. • If V has odd dimension then CI (V,Q) is a central simple algebra over K. The structure of Clifford algebras can be worked out explicitly using the following result. Suppose that U has even dimension and a non-singular bilinear form with discriminant d, and suppose that V is another vector space with a quadratic form. The Clifford algebra of U+V is isomorphic to the tensor product of the Clifford algebras of U and (-1) dV, which is the space V with its quadratic form multiplied by (-1) d. Over the reals, this implies in particular that Cl p+ 2, q (R) - M 2 (R)®C7 g , p (i Cl p+ i, q+ i(M) = M 2 (R)®C* p , g dp,q+2(M) ~ H®C7 giP (R). These formulas can be used to find the structure of all real Clifford algebras; see the classification of Clifford algebras. Notably, the Morita equivalence class of a Clifford algebra (its representation theory: the equivalence class of the category of modules over it) depends only on the signature p — qmod 8. This is an algebraic form of Bott periodicity. Clifford algebra 136 The Clifford group T In this section we assume that Vis finite dimensional and the quadratic form Q is nondegenerate. The invertible elements of the Clifford algebra act on it by twisted conjugation: conjugation by x maps y \— > xya(x)~ ■ The Clifford group Y is defined to be the set of invertible elements x that stabilize vectors, meaning that xva,[x)~ £ V for all v in V. This formula also defines an action of the Clifford group on the vector space V that preserves the norm Q, and so gives a homomorphism from the Clifford group to the orthogonal group. The Clifford group contains all elements r of V of nonzero norm, and these act on V by the corresponding reflections that take v to v - <v,r>r/Q(r) (In characteristic 2 these are called orthogonal transvections rather than reflections.) The Clifford group T is the disjoint union of two subsets T and T , where r' is the subset of elements of degree i. The subset T is a subgroup of index 2 in T. If V is a finite dimensional real vector space with positive definite (or negative definite) quadratic form then the Clifford group maps onto the orthogonal group of V with respect to the form (by the Cartan-Dieudonne theorem) and the kernel consists of the nonzero elements of the field K. This leads to exact sequences 1 -> K* -> T -»■ Ov(K) -> 1, i -> k* -> r° -> SO v (K) -> 1. Over other fields or with indefinite forms, the map is not in general onto, and the failure is captured by the spinor Spinor norm In arbitrary characteristic, the spinor norm Q is defined on the Clifford group by Q(x) = x t x. It is a homomorphism from the Clifford group to the group K of non-zero elements of K. It coincides with the quadratic form Q of V when V is identified with a subspace of the Clifford algebra. Several authors define the spinor norm slightly differently, so that it differs from the one here by a factor of -1, 2, or -2 on T . The difference is not very important in characteristic other than 2. The nonzero elements of K have spinor norm in the group K of squares of nonzero elements of the field K. So when V is finite dimensional and non-singular we get an induced map from the orthogonal group of V to the group K IK , also called the spinor norm. The spinor norm of the reflection of a vector r has image Q(r) in K IK , and this property uniquely defines it on the orthogonal group. This gives exact sequences: 1 -> {±1} -> Pm v (K) -> O v (K) ->■ K*/K* 2 , 1 -> {±1} -► Spirv(X) -> SO v (K) -> K*/K* 2 . Note that in characteristic 2 the group {±1 } has just one element. From the point of view of Galois cohomology of algebraic groups, the spinor norm is a connecting homomorphism on cohomology. Writing \i for the algebraic group of square roots of 1 (over a field of characteristic not 2 it is roughly the same as a two-element group with trivial Galois action), the short exact sequence 1 — s- (i 2 — * Piny — > Oy — > 1 yields a long exact sequence on cohomology, which begins 1 -► H°(ji2; K) -► H°(Pm v ; K) -> H°{O v ; K) -► ff 1 ^; X). Clifford algebra 137 The Oth Galois cohomology group of an algebraic group with coefficients in K is just the group of ^-valued points: H°(G; K) = G(K), and H 1 (fi2', K) = K* I K* 2 , which recovers the previous sequence 1 -> {±1} -> Pm v (K) -> O^if) -»• ir/ir 2 , where the spinor norm is the connecting homomorphism H°(Oy' K) — > H 1 !^' K). Spin and Pin groups In this section we assume that V is finite dimensional and its bilinear form is non-singular. (If K has characteristic 2 this implies that the dimension of V is even.) The Pin group Pin (K) is the subgroup of the Clifford group Y of elements of spinor norm 1, and similarly the Spin group Spin (K) is the subgroup of elements of Dickson invariant in Pin (K). When the characteristic is not 2, these are the elements of determinant 1. The Spin group usually has index 2 in the Pin group. Recall from the previous section that there is a homomorphism from the Clifford group onto the orthogonal group. We define the special orthogonal group to be the image of T . If K does not have characteristic 2 this is just the group of elements of the orthogonal group of determinant 1 . If K does have characteristic 2, then all elements of the orthogonal group have determinant 1, and the special orthogonal group is the set of elements of Dickson invariant 0. There is a homomorphism from the Pin group to the orthogonal group. The image consists of the elements of spinor norm 1 G K IK . The kernel consists of the elements +1 and -1, and has order 2 unless K has characteristic 2. Similarly there is a homomorphism from the Spin group to the special orthogonal group of V. In the common case when V is a positive or negative definite space over the reals, the spin group maps onto the special orthogonal group, and is simply connected when V has dimension at least 3. Further the kernel of this homomorphism consists of 1 and -l.So in this case the spin group, Spin(n), is a double cover of SO(n). Please note, however, that the simple connectedness of the spin group is not true in general: if V is R for p and q both at least 2 then the spin group is not simply connected. In this case the algebraic group Spin is simply connected as an algebraic group, even though its group of real valued points Spin (R) is not simply connected. This is a rather subtle point, which completely confused the authors of at least one standard book about spin groups. Spinors Clifford algebras CD (C), with p+q=2n even, are matrix algebras which have a complex representation of dimension 2 . By restricting to the group Pin (R) we get a complex representation of the Pin group of the same P>4 dimension, called the spin representation. If we restrict this to the spin group Spin (R) then it splits as the sum of two half spin representations (or Weyl representations) of dimension 2 ,»-! M If p+q=2n+l is odd then the Clifford algebra CD (C) is a sum of two matrix algebras, each of which has a n ' p ' q representation of dimension 2 , and these are also both representations of the Pin group Pin (R). On restriction to the spin group Spin (R) these become isomorphic, so the spin group has a complex spinor representation of n M dimension 2 . More generally, spinor groups and pin groups over any field have similar representations whose exact structure depends on the structure of the corresponding Clifford algebras: whenever a Clifford algebra has a factor that is a matrix algebra over some division algebra, we get a corresponding representation of the pin and spin groups over that division algebra. For examples over the reals see the article on spinors. Clifford algebra 138 Real spinors To describe the real spin representations, one must know how the spin group sits inside its Clifford algebra. The Pin group, Pin is the set of invertible elements in CI which can be written as a product of unit vectors: p.q p.q Pin Ptq = {v x v 2 ...v r \Vi, \\vi\\ = ±1}. Comparing with the above concrete realizations of the Clifford algebras, the Pin group corresponds to the products of arbitrarily many reflections: it is a cover of the full orthogonal group 0{p,q). The Spin group consists of those elements of Pin which are products of an even number of unit vectors. Thus by the Cartan-Dieudonne theorem PA Spin is a cover of the group of proper rotations SO{p,q). Let a : CI — > CI be the automorphism which is given by -Id acting on pure vectors. Then in particular, Spin is the subgroup of Pin whose elements are fixed by a. Let PA Cl° P!q = {x€Cl p Ja{x)=x}. (These are precisely the elements of even degree in CI .) Then the spin group lies within CI The irreducible representations of CI restrict to give representations of the pin group. Conversely, since the pin PA group is generated by unit vectors, all of its irreducible representation are induced in this manner. Thus the two representations coincide. For the same reasons, the irreducible representations of the spin coincide with the irreducible representations of CI p.q To classify the pin representations, one need only appeal to the classification of Clifford algebras. To find the spin representations (which are representations of the even subalgebra), one can first make use of either of the isomorphisms (see above) „o p.q p.q- Cr a C£ , for q > J3 p.q q.p- and realize a spin representation in signature (p,q) as a pin representation in either signature (p,q-l) or (q,p-l). Cr =C/ „forp>0 p-l Applications Differential geometry One of the principal applications of the exterior algebra is in differential geometry where it is used to define the bundle of differential forms on a smooth manifold. In the case of a (pseudo-)Riemannian manifold, the tangent spaces come equipped with a natural quadratic form induced by the metric. Thus, one can define a Clifford bundle in analogy with the exterior bundle. This has a number of important applications in Riemannian geometry. Perhaps more importantly is the link to a spin manifold, its associated spinor bundle and spin manifolds. Physics Clifford algebras have numerous important applications in physics. Physicists usually consider a Clifford algebra to be an algebra spanned by matrices y ,...,y, called Dirac matrices which have the property that lilj + Ijli = 2 Vij where r| is the matrix of a quadratic form of signature (1,3). These are exactly the defining relations for the Clifford algebra CI (C) (up to an unimportant factor of 2), which by the classification of Clifford algebras is isomorphic to the algebra of 4 by 4 complex matrices. The Dirac matrices were first written down by Paul Dirac when he was trying to write a relativistic first-order wave equation for the electron, and give an explicit isomorphism from the Clifford algebra to the algebra of complex matrices. The result was used to define the Dirac equation and introduce the Dirac operator. The entire Clifford algebra shows up in quantum field theory in the form of Dirac field bilinears. Clifford algebra 139 Computer Vision Recently, Clifford algebras have been applied in the problem of action recognition and classification in computer vision. Rodriguez et al. propose a Clifford embedding to generalize traditional MACH filters to video (3D spatiotemporal volume), and vector-valued data such as optical flow. Vector-valued data is analyzed using the Clifford Fourier transform. Based on these vectors action filters are synthesized in the Clifford Fourier domain and recognition of actions is performed using Clifford Correlation. The authors demonstrate the effectiveness of the Clifford embedding by recognizing actions typically performed in classic feature films and sports broadcast television. Notes [1] Mathematicians who work with real Clifford algebras and prefer positive definite quadratic forms (especially those working in index theory) 2 sometimes use a different choice of sign in the fundamental Clifford identity. That is, they take v = -Q(v). One must replace Q with —Q in going from one convention to the other. [2] Thus the group algebra K[Z/2] is semisimple and the Clifford algebra splits into eigenspaces of the main involution. [3] We are still assuming that the characteristic is not 2. [4] The opposite is true when using the alternate (-) sign convention for Clifford algebras: it is the conjugate which is more important. In general, the meanings of conjugation and transpose are interchanged when passing from one sign convention to the other. For example, in the convention used here the inverse of a vector is given by y = y / Q(v) while in the (-) convention it is given by v- 1 =v/Q{v). [5] Rodriguez, Mikel; Shah, M (2008). "Action MACH: A Spatio-Temporal Maximum Average Correlation Height Filter for Action Classification". Computer Vision and Pattern Recognition (CVPR). References • Bourbaki, Nicolas (1988), Algebra, Berlin, New York: Springer-Verlag, ISBN 978-3-540-19373-9, section XI.9. • Carnahan, S. Borcherds Seminar Notes, Uncut. Week 5, "Spinors and Clifford Algebras". • Lawson, H. Blaine; Michelsohn, Marie-Louise (1989), Spin Geometry, Princeton, NJ: Princeton University Press, ISBN 978-0-691-08542-5. An advanced textbook on Clifford algebras and their applications to differential geometry. • Lounesto, Pertti (2001), Clifford algebras and spinors, Cambridge: Cambridge University Press, ISBN 978-0-521-00551-7 • Porteous, Ian R. (1995), Clifford algebras and the classical groups, Cambridge: Cambridge University Press, ISBN 978-0-521-55177-9 External links • Planetmath entry on Clifford algebras (http://planetmath.org/encyclopedia/CliffordAlgebra2.html) • A history of Clifford algebras (http://members.fortunecity.com/jonhays/clifhistory.htm) (unverified) • John Baez on Clifford algebras (http://www.math.ucr.edu/home/baez/octonions/node6.html) Von Neumann algebra 140 Von Neumann algebra In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by the study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. Two basic examples of von Neumann algebras are as follows. The ring L°°(R) of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, which acts by pointwise multiplication on the 2 Hilbert space L (R) of square integrable functions. The algebra B(H) of all bounded operators on a Hilbert space H is a von Neumann algebra, non-commutative if the Hilbert space has dimension at least 2. Von Neumann algebras were first studied by von Neumann (1929); he and Francis Murray developed the basic theory, under the original name of rings of operators, in a series of papers written in the 1930s and 1940s (F.J. Murray & J. von Neumann 1936, 1937, 1943; J. von Neumann 1938, 1940, 1943, 1949), reprinted in the collected works of von Neumann (1961). Introductory accounts of von Neumann algebras are given in the online notes of Jones (2003) and Wassermann (1991) and the books by Dixmier (1981), Schwartz (1967), Blackadar (2005) and Sakai (1971). The three volume work by Takesaki (1979) gives an encyclopedic account of the theory. The book by Connes (1994) discusses more advanced topics. Definitions There are three common ways to define von Neumann algebras. The first and most common way is to define them as weakly closed * algebras of bounded operators (on a Hilbert space) containing the identity. In this definition the weak (operator) topology can be replaced by many other common topologies including the strong, ultrastrong or ultraweak operator topologies. The *-algebras of bounded operators that are closed in the norm topology are C*-algebras, so in particular any von Neumann algebra is a C*-algebra. The second definition is that a von Neumann algebra is a subset of the bounded operators closed under * and equal to its double commutant, or equivalently the commutant of some subset closed under *. The von Neumann double commutant theorem (von Neumann 1929) says that the first two definitions are equivalent. The first two definitions describe a von Neumann algebras concretely as a set of operators acting on some given Hilbert space. Sakai (1971) showed that von Neumann algebras can also be defined abstractly as C*-algebras that have a predual; in other words the von Neumann algebra, considered as a Banach space, is the dual of some other Banach space called the predual. The predual of a von Neumann algebra is in fact unique up to isomorphism. Some authors use "von Neumann algebra" for the algebras together with a Hilbert space action, and "W*-algebra" for the abstract concept, so a von Neumann algebra is a W*-algebra together with a Hilbert space and a suitable faithful unital action on the Hilbert space. The concrete and abstract definitions of a von Neumann algebra are similar to the concrete and abstract definitions of a C*-algebra, which can be defined either as norm-closed * algebras of operators on a Hilbert space, or as Banach *-algebras such that \\a a*ll=llall lb*ll. Von Neumann algebra 141 Terminology Some of the terminology in von Neumann algebra theory can be confusing, and the terms often have different meanings outside the subject. • A factor is a von Neumann algebra with trivial center, i.e. a center consisting only of scalar operators. • A finite von Neumann algebra is one which is the direct integral of finite factors. Similarly, properly infinite von Neumann algebras are the direct integral of properly infinite factors. • A von Neumann algebra that acts on a separable Hilbert space is called separable. Note that such algebras are rarely separable in the norm topology. • The von Neumann algebra generated by a set of bounded operators on a Hilbert space is the smallest von Neumann algebra containing all those operators. • The tensor product of two von Neumann algebras acting on two Hilbert spaces is defined to be the von Neumann algebra generated by their algebraic tensor product, considered as operators on the Hilbert space tensor product of the Hilbert spaces. By forgetting about the topology on a von Neumann algebra, we can consider it a (unital) *-algebra, or just a ring. Von Neumann algebras are semihereditary: every finitely generated submodule of a projective module is itself projective. There have been several attempts to axiomatize the underlying rings of von Neumann algebras, including Baer *-rings and AW* algebras. The *-algebra of affiliated operators of a finite von Neumann algebra is a von Neumann regular ring. (The von Neumann algebra itself is in general not von Neumann regular.) Commutative von Neumann algebras Main article: Abelian von Neumann algebra The relationship between commutative von Neumann algebras and measure spaces is analogous to that between commutative C*-algebras and locally compact Hausdorff spaces. Every commutative von Neumann algebra is isomorphic to L°°{X) for some measure space (X, \i) and conversely, for every o-finite measure space X, the * algebra L°°(X) is a von Neumann algebra. Due to this analogy, the theory of von Neumann algebras has been called noncommutative measure theory, while the theory of C*-algebras is sometimes called noncommutative topology (Connes 1994). Projections Operators £ in a von Neumann algebra for which E = EE = E* are called projections; they are exactly the operators which give an orthogonal projection of H onto some closed subspace. A subspace of the Hilbert space H is said to belong to the von Neumann algebra M if it is the image of some projection in M. Informally these are the closed subspaces that can be described using elements of M, or that M "knows" about. The closure of the image of any operator in M, or the kernel of any operator in M belong to M, and the closure of the image of any subspace belonging to M under an operator of M also belongs to M. There is a 1:1 correspondence between projections of M and subspaces that belong to it. The basic theory of projections was worked out by Murray & von Neumann (1936). Two subspaces belonging to M are called (Murray-von Neumann) equivalent if there is a partial isometry mapping the first isomorphically onto the other that is an element of the von Neumann algebra (informally, if M "knows" that the subspaces are isomorphic). This induces a natural equivalence relation on projections by defining E to be equivalent to F if the corresponding subspaces are equivalent, or in other words if there is a partial isometry of H that maps the image of E isometrically to the image of F and is an element of the von Neumann algebra. Another way of stating this is that E is equivalent to F if E=uu and F=u u for some partial isometry u in M. The equivalence relation ~ thus defined is additive in the following sense: Suppose E ~ F and E ~ F . If E L E and F L F , then E + E ~ F + F . This is not true in general if one requires unitary equivalence in the definition of Von Neumann algebra 142 ~, i.e. if we say E is equivalent to F if u*Eu = F for some unitary u. . The subspaces belonging to M are partially ordered by inclusion, and this induces a partial order < of projections. There is also a natural partial order on the set of equivalence classes of projections, induced by the partial order < of projections. If Mis a factor, < is a total order on equivalence classes of projections, described in the section on traces below. A projection (or subspace belonging to M) E is said to be finite if there is no projection F < E that is equivalent to E. For example, all finite-dimensional projections (or subspaces) are finite (since isometries between Hilbert spaces leave the dimension fixed), but the identity operator on an infinite-dimensional Hilbert space is not finite in the von Neumann algebra of all bounded operators on it, since it is isometrically isomorphic to a proper subset of itself. However it is possible for infinite dimensional subspaces to be finite. Orthogonal projections are noncommutative analogues of indicator functions in L°°(R). L°°(R) is the 11-11 -closure of the subspace generated by the indicator functions. Similarly, a von Neumann algebra is generated by its projections; this is a consequence of the spectral theorem for self-adjoint operators. Factors A von Neumann algebra N whose center consists only of multiples of the identity operator is called a factor, von Neumann (1949) showed that every von Neumann algebra on a separable Hilbert space is isomorphic to a direct integral of factors. This decomposition is essentially unique. Thus, the problem of classifying isomorphism classes of von Neumann algebras on separable Hilbert spaces can be reduced to that of classifying isomorphism classes of factors. Murray & von Neumann (1936) showed that every factor has one of 3 types as described below. The type classification can be extended to von Neumann algebras that are not factors, and a von Neumann algebra is of type X if it can be decomposed as a direct integral of type X factors; for example, every commutative von Neumann algebra has type I . Every von Neumann algebra can be written uniquely as a sum of von Neumann algebras of types I, II, and III. There are several other ways to divide factors into classes that are sometimes used: • A factor is called discrete (or occasionally tame) if it has type I, and continuous (or occasionally wild) if it has type II or III. • A factor is called semifinite if it has type I or II, and purely infinite if it has type III. • A factor is called finite if the projection 1 is finite and properly infinite otherwise. Factors of types I and II may be either finite or properly infinite, but factors of type III are always properly infinite. Type I factors A factor is said to be of type I if there is a minimal projection E * 0, i.e. a projection E such that there is no other projection F with < F < E. Any factor of type I is isomorphic to the von Neumann algebra of all bounded operators on some Hilbert space; since there is one Hilbert space for every cardinal number, isomorphism classes of factors of type I correspond exactly to the cardinal numbers. Since many authors consider von Neumann algebras only on separable Hilbert spaces, it is customary to call the bounded operators on a Hilbert space of finite dimension n a factor of type I , and the bounded operators on a separable infinite-dimensional Hilbert space, a factor of type I . Von Neumann algebra 143 Type II factors A factor is said to be of type II if there are no minimal projections but there are non-zero finite projections. This implies that every projection E can be halved in the sense that there are equivalent projections F and G such that E = F + G. If the identity operator in a type II factor is finite, the factor is said to be of type II ; otherwise, it is said to be of type II . The best understood factors of type II are the hyperfinite type II factor and the hyperfinite type II factor, found by Murray & von Neumann (1936). These are the unique hyperfinite factors of types II and II ; there 1 oo are an uncountable number of other factors of these types that are the subject of intensive study. Murray & von Neumann (1937) proved the fundamental result that a factor of type II has a unique finite tracial state, and the set of traces of projections is [0,1]. A factor of type II has a semifinite trace, unique up to rescaling, and the set of traces of projections is [0,°°]. The set of real numbers X such that there is an automorphism rescaling the trace by a factor of X is called the fundamental group of the type II factor. The tensor product of a factor of type II and an infinite type I factor has type II , and conversely any factor of type II can be constructed like this. The fundamental group of a type II factor is defined to be the fundamental group of its tensor product with the infinite (separable) factor of type I. For many years it was an open problem to find a type II factor whose fundamental group was not the group of all positive reals, but Connes then showed that the von Neumann group algebra of a countable discrete group with Kazhdan's property T (the trivial representation is isolated in the dual space), such as SL (Z), has a countable fundamental group. Subsequently Sorin Popa showed 2 that the fundamental group can be trivial for certain groups, including the semidirect product of Z by SL (Z). An example of a type II factor is the von Neumann group algebra of a countable infinite discrete group such that every non-trivial conjugacy class is infinite. McDuff (1969) found an uncountable family of such groups with non-isomorphic von Neumann group algebras, thus showing the existence of uncountably many different separable type II factors. Type III factors Lastly, type III factors are factors that do not contain any nonzero finite projections at all. In their first paper Murray & von Neumann (1936) were unable to decide whether or not they existed; the first examples were later found by von Neumann (1940). Since the identity operator is always infinite in those factors, they were sometimes called type III in the past, but recently that notation has been superseded by the notation III , where X is a real number in the interval [0,1]. More precisely, if the Connes spectrum (of its modular group) is 1 then the factor is of type III , if the Connes spectrum is all integral powers of X for < X < 1, then the type is III , and if the Connes spectrum is all A. positive reals then the type is III . (The Connes spectrum is a closed subgroup of the positive reals, so these are the only possibilities.) The only trace on type III factors takes value °° on all non-zero positive elements, and any two non-zero projections are equivalent. At one time type III factors were considered to be intractable objects, but Tomita— Takesaki theory has led to a good structure theory. In particular, any type III factor can be written in a canonical way as the crossed product of a type II factor and the real numbers. Von Neumann algebra 144 The predual Any von Neumann algebra M has a predual M , which is the Banach space of all ultraweakly continuous linear functionals on M. As the name suggests, M is (as a Banach space) the dual of its predual. The predual is unique in the sense that any other Banach space whose dual is M is canonically isomorphic to M . Sakai (1971) showed that the existence of a predual characterizes von Neumann algebras among C* algebras. The definition of the predual given above seems to depend on the choice of Hilbert space that M acts on, as this determines the ultraweak topology. However the predual can also be defined without using the Hilbert space that M acts on, by defining it to be the space generated by all positive normal linear functionals on M. (Here "normal" means that it preserves suprema when applied to increasing nets of self adjoint operators; or equivalently to increasing sequences of projections.) The predual M^ is a closed subspace of the dual M (which consists of all norm-continuous linear functionals on M) but is generally smaller. The proof that M t is (usually) not the same as M is nonconstructive and uses the axiom of choice in an essential way; it is very hard to exhibit explicit elements of M that are not in M^. For example, exotic positive linear forms on the von Neumann algebra f° (Z) are given by free ultrafilters; they correspond to exotic *-homomorphisms into C and describe the Stone-Cech compactification of Z. Examples: 1 . The predual of the von Neumann algebra L°°(R) of essentially bounded functions on R is the Banach space L (R) of integrable functions. The dual of L°°(R) is strictly larger than L (R) For example, a functional on L°°(R) that extends the Dirac measure 6 on the closed subspace of bounded continuous functions C (R) cannot be represented as a function in L (R). 2. The predual of the von Neumann algebra B{H) of bounded operators on a Hilbert space H is the Banach space of all trace class operators with the trace norm IIAII= Tr(IAI). The Banach space of trace class operators is itself the dual of the C*-algebra of compact operators (which is not a von Neumann algebra). Weights, states, and traces Weights and their special cases states and traces are discussed in detail in (Takesaki 1979). • A weight to on a von Neumann algebra is a linear map from the set of positive elements (those of the form a a) to [0,°°]- • A positive linear functional is a weight with (o(l) finite (or rather the extension of a> to the whole algebra by linearity). • A state is a weight with co(l)=l. • A trace is a weight with w(aa )-w(a a) for all a. • A tracial state is a trace with to(l)=l. Any factor has a trace such that the trace of a non-zero projection is non-zero and the trace of a projection is infinite if and only if the projection is infinite. Such a trace is unique up to rescaling. For factors that are separable or finite, two projections are equivalent if and only if they have the same trace. The type of a factor can be read off from the possible values of this trace as follows: • Type I : 0, x, 2x, ....,nx for some positive x (usually normalized to be \ln or 1). • Type I : 0, x, 2x, ....,°° for some positive x (usually normalized to be 1). • Type II : [0,x] for some positive x (usually normalized to be 1). • Type II j [OH- • Type III: 0,°°. If a von Neumann algebra acts on a Hilbert space containing a norm 1 vector v, then the functional a — > (av,v) is a normal state. This construction can be reversed to give an action on a Hilbert space from a normal state: this is the Von Neumann algebra 145 GNS construction for normal states. Modules over a factor Given an abstract separable factor, one can ask for a classification of its modules, meaning the separable Hilbert spaces that it acts on. The answer is given as follows: every such module H can be given an M-dimension dim M (H) (not its dimension as a complex vector space) such that modules are isomorphic if and only if they have the same M-dimension. The M-dimension is additive, and a module is isomorphic to a subspace of another module if and only if it has smaller or equal M-dimension. A module is called standard if it has a cyclic separating vector. Each factor has a standard representation, which is unique up to isomorphism. The standard representation has an antilinear involution J such that JMJ = M'. For finite factors the standard module is given by the GNS construction applied to the unique normal tracial state and the M-dimension is normalized so that the standard module has M-dimension 1, while for infinite factors the standard module is the module with M-dimension equal to °°. The possible M-dimensions of modules are given as follows: • Type I (« finite): The M-dimension can be any of 0/«, l/«, 2/«, 3/«, ..., °°. The standard module has M-dimension " 2 1 (and complex dimension n .) • Type I The M-dimension can be any of 0, 1, 2, 3, ..., °°. The standard representation of B{H) is H®H; its M-dimension is °°. • Type II : The M-dimension can be anything in [0, °°], It is normalized so that the standard module has M-dimension 1 . The M-dimension is also called the coupling constant of the module H. • Type II : The M-dimension can be anything in [0, <*>]. There is in general no canonical way to normalize it; the factor may have outer automorphisms multiplying the M-dimension by constants. The standard representation is the one with M-dimension °°. • Type III: The M-dimension can be or °°. Any two non-zero modules are isomorphic, and all non-zero modules are standard. Amenable von Neumann algebras Connes (1976) and others proved that the following conditions on a von Neumann algebra M on a separable Hilbert space H are all equivalent: • M is hyperfinite or AFD or approximately finite dimensional or approximately finite: this means the algebra contains an ascending sequence of finite dimensional subalgebras with dense union. (Warning: some authors use "hyperfinite" to mean "AFD and finite".) • M is amenable: this means that the derivations of M with values in a normal dual Banach bimodule are all inner. • M has Schwartz's property P: for any bounded operator TonH the weak operator closed convex hull of the elements uTu contains an element commuting with M. • M is semidiscrete: this means the identity map from M to M is a weak pointwise limit of completely positive maps of finite rank. • M has property E or the Hakeda-Tomiyama extension property: this means that there is a projection of norm 1 from bounded operators on H to M '. • M is injective: any completely positive linear map from any self adjoint closed subspace containing 1 of any unital C -algebra A to M can be extended to a completely positive map from A to M. There is no generally accepted term for the class of algebras above; Connes has suggested that amenable should be the standard term. Von Neumann algebra 146 The amenable factors have been classified: there is a unique one of each of the types 1,1 , II, , II , III, , for 0<X< 1, n ~ 1 ~ A. and the ones of type III correspond to certain ergodic flows. (For type III calling this a classification is a little misleading, as it is known that there is no easy way to classify the corresponding ergodic flows.) The ones of type I and II were classified by Murray & von Neumann (1943), and the remaining ones were classified by Connes (1976), except for the type III case which was completed by Haagerup. All amenable factors can be constructed using the group-measure space construction of Murray and von Neumann for a single ergodic transformation. In fact they are precisely the factors arising as crossed products by free ergodic actions of Z or Z on abelian von Neumann algebras L°°(X). Type I factors occur when the measure space X is atomic and the action transitive. When X is diffuse or non-atomic, it is equivalent to [0,1] as a measure space. Type II factors occur when X admits an equivalent finite (II ) or infinite (II ) measure, invariant under Z ■ Type III factors occur in the remaining cases where there is no invariant measure, but only an invariant measure class: these factors are called Krieger factors. Tensor products of von Neumann algebras The Hilbert space tensor product of two Hilbert spaces is the completion of their algebraic tensor product. One can define a tensor product of von Neumann algebras (a completion of the algebraic tensor product of the algebras considered as rings), which is again a von Neumann algebra, and act on the tensor product of the corresponding Hilbert spaces. The tensor product of two finite algebras is finite, and the tensor product of an infinite algebra and a non-zero algebra is infinite. The type of the tensor product of two von Neumann algebras (I, II, or III) is the maximum of their types. The commutation theorem for tensor products states that (M ® N)' = M'®N' (where M' denotes the commutant of M). The tensor product of an infinite number of von Neumann algebras, if done naively, is usually a ridiculously large non-separable algebra. Instead von Neumann (1938) showed that one should choose a state on each of the von Neumann algebras, use this to define a state on the algebraic tensor product, which can be used to product a Hilbert space and a (reasonably small) von Neumann algebra. Araki & Woods (1968) studied the case where all the factors are finite matrix algebras; these factors are called Araki-Woods factors or ITPFI factors (ITPFI stands for "infinite tensor product of finite type I factors"). The type of the infinite tensor product can vary dramatically as the states are changed; for example, the infinite tensor product of an infinite number of type I factors can have any type depending on the choice of states. In particular Powers (1967) found an uncountable family of non-isomorphic hyperfinite type III factors for 0<X<1, called Powers factors, by taking an infinite tensor product of type I factors, each with the state given by : x i — ► Tr ( "^^T 1 ^ 1 x. V ° ATI/ All hyperfinite von Neumann algebras not of type III are isomorphic to Araki-Woods factors, but there are uncountably many of type III that are not. Bimodules and subfactors A bimodule (or correspondence) is a Hilbert space H with module actions of two commuting von Neumann algebras. Bimodules have a much richer structure than that of modules. Any bimodule over two factors always gives a subfactor since one of the factors is always contained in the commutant of the other. There is also a subtle relative tensor product operation due to Connes on bimodules. The theory of subfactors, initiated by Vaughan Jones, reconciles these two seemingly different points of view. Bimodules are also important for the von Neumann group algebra M of a discrete group p. Indeed if V is any unitary representation of p, then, regarding pas the diagonal subgroup of px p, the corresponding induced 2 representation on / ( P,V) is naturally a bimodule for two commuting copies of M. Important representation Von Neumann algebra 147 theoretic properties of pcan be formulated entirely in terms of bimodules and therefore make sense for the von Neumann algebra itself. For example Connes and Jones gave a definition of an analogue of Kazhdan's Property T for von Neumann algebras in this way. Non-amenable factors Von Neumann algebras of type I are always amenable, but for the other types there are an uncountable number of different non-amenable factors, which seem very hard to classify, or even distinguish from each other. Nevertheless Voiculescu has shown that the class of non-amenable factors coming from the group-measure space construction is disjoint from the class coming from group von Neumann algebras of free groups. Later Narutaka Ozawa proved that group von Neumann algebras of hyperbolic groups yield prime type II factors, i.e. ones that cannot be factored as tensor products of type II factors, a result first proved by Leeming Ge for free group factors using Voiculescu's free entropy. Popa's work on fundamental groups of non-amenable factors represents another significant advance. The theory of factors "beyond the hyperfinite" is rapidly expanding at present, with many new and surprising results; it has close links with rigidity phenomena in geometric group theory and ergodic theory. Examples • The essentially bounded functions on a o-finite measure space form a commutative (type I ) von Neumann 2 algebra acting on the L functions. For certain non-o-finite measure spaces, usually considered pathological, L°°(X) is not a von Neumann algebra; for example, the o-algebra of measurable sets might be the countable-cocountable algebra on an uncountable set. • The bounded operators on any Hilbert space form a von Neumann algebra, indeed a factor, of type I. • If we have any unitary representation of a group G on a Hilbert space H then the bounded operators commuting with G form a von Neumann algebra G', whose projections correspond exactly to the closed subspaces of H invariant under G. Equivalent subrepresentations correspond to equivalent projections in G'. The double commutant G" of G is also a von Neumann algebra. 2 • The von Neumann group algebra of a discrete group G is the algebra of all bounded operators on H = I (G) commuting with the action of G on H through right multiplication. One can show that this is the von Neumann algebra generated by the operators corresponding to multiplication from the left with an element g G G. It is a factor (of type II ) if every non-trivial conjugacy class of G is infinite (for example, a non-abelian free group), and is the hyperfinite factor of type II if in addition G is a union of finite subgroups (for example, the group of all permutations of the integers fixing all but a finite number of elements). • The tensor product of two von Neumann algebras, or of a countable number with states, is a von Neumann algebra as described in the section above. • The crossed product of a von Neumann algebra by a discrete (or more generally locally compact) group can be defined, and is a von Neumann algebra. Special cases are the group-measure space construction of Murray and von Neumann and Krieger factors. • The von Neumann algebras of a measurable equivalence relation and a measurable groupoid can be defined. These examples generalise von Neumann group algebras and the group-measure space construction. Von Neumann algebra 148 Applications Von Neumann algebras have found applications in diverse areas of mathematics like knot theory, statistical mechanics, Quantum field theory, Local quantum physics, Free probability, Noncommutative geometry, representation theory, geometry, and probability. References • Araki, H.; Woods, E. J. (1968), "A classification of factors", Publ. Res. Inst. Math. Sci. Ser. A 4: 51-130, doi:10.2977/prims/1195195263MR0244773 • Blackadar, B. (2005), Operator algebras, Springer, ISBN 3-540-28486-9 • Connes, A. (1976), "Classification of Injective Factors" , The Annals of Mathematics 2nd Ser. 104 (1): 73—1 15, doi: 10.2307/197 1057 • Connes, A. (1994), Non-commutative geometry , Academic Press, ISBN 0-12-185860-X. • Dixmier, J. (1981), Von Neumann algebras, ISBN 0-444-86308-7 (A translation of Dixmier, J. (1957), Les algebres d'operateurs dans Vespace hilbertien: algebres de von Neumann, Gauthier-Villars, the first book about von Neumann algebras.) • Jones, V.F.R. (2003), von Neumann algebras ; incomplete notes from a course. • McDuff, Dusa (1969), "Uncountably many II factors" [4] , Ann of Math. (Annals of Mathematics) 90 (2): 372-377, doi: 10.2307/1970730 • Murray, F. J., "The rings of operators papers", The legacy of John von Neumann (Hempstead, NY, 1988), Proc. Sympos. Pure Math., 50, Providence, RL: Amer. Math. Soc, pp. 57-60, ISBN 0-8218-4219-6 A historical account of the discovery of von Neumann algebras. • Murray, F.J.; von Neumann, J. (1936), "On rings of operators" , Ann. Of Math. (2) (Annals of Mathematics) 37 (1): 1 16—229, doi: 10.2307/1968693. This paper gives their basic properties and the division into types I, II, and III, and in particular finds factors not of type I. • Murray, F.J.; von Neumann, J. (1937), "On rings of operators II" , Trans. Amer. Math. Soc. (American Mathematical Society) 41 (2): 208-248, doi: 10.2307/1989620. This is a continuation of the previous paper, that studies properties of the trace of a factor. • Murray, F.J.; von Neumann, J. (1943), "On rings of operators IV" , Ann. Of Math. (2) (Annals of Mathematics) 44 (4): 716—808, doi:10. 2307/1969107. This studies when factors are isomorphic, and in particular shows that all approximately finite factors of type II are isomorphic. • Powers, Robert T. (1967), "Representations of Uniformly Hyperfinite Algebras and Their Associated von Neumann Rings" [8] , The Annals of Mathematics, 2nd Ser. 86 (1): 138-171, doi: 10.2307/1970364 • Sakai, S. (1971), C*-algebras and W*-algebras, Springer, ISBN 3-540-63633-1 • Schwartz, Jacob (1967), W-* Algebras, ISBN 0-677-00670-5 • Shtern, A.I. (2001), "von Neumann algebra" , in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104 • Takesaki, M. (1979), Theory of Operator Algebras I, II, III, ISBN 3-540-42248-X ISBN 3-540-42914-X ISBN 3-540-42913-1 • von Neumann, J. (1929), "Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren", Math. Ann. 102: 370—427, doi:10.1007/BF01782352. The original paper on von Neumann algebras. • von Neumann, J. (1936), "On a Certain Topology for Rings of Operators" , The Annals of Mathematics 2nd Ser. 37 (1): 111-115, doi: 10.2307/1968692. This defines the ultrastrong topology, von Neumann, J. (1938), "On infinite direct products" , > products of Hilbert spaces and the algebras acting on them. ri2i von Neumann, J. (1940), "On rings of operators III" , At 94-161, doi: 10.2307/1968823. This shows the existence of factors of type III. von Neumann, J. (1938), "On infinite direct products" , Compos. Math. 6: 1—77. This discusses infinite tensor on the von Neumann, J. (1940), "On rings of operators III" , Ann. Of Math. (2) (Annals of Mathematics) 41 (1): Von Neumann algebra 149 ri3i • von Neumann, J. (1943), "On Some Algebraical Properties of Operator Rings" , The Annals of Mathematics 2nd Ser. 44 (4): 709—715, doi: 10.2307/1969106. This shows that some apparently topological properties in von Neumann algebras can be defined purely algebraically. ri4i • von Neumann, J. (1949), "On Rings of Operators. Reduction Theory" , The Annals of Mathematics 2nd Ser. 50 (2): 401—485, doi: 10.2307/1969463. This discusses how to write a von Neumann algebra as a sum or integral of factors. • von Neumann, John (1961), Taub, A.H., ed., Collected Works, Volume III: Rings of Operators, NY: Pergamon Press. Reprints von Neumann's papers on von Neumann algebras. • Wassermann, A. J. (1991), Operators on Hilbert space References [I] http://links.jstor.org/sici?sici=0003-486X%28197607%292%3A104%3Al%3C73%3ACOIFC%3E2.0.CO%3B2-V [2] http://www.alainconnes.org/docs/book94bigpdf.pdf [3] http://www.math.berkeley.edu/~vfr/MATH20909/VonNeumann2009.pdf [4] http://links.jstor.org/sici?sici=0003-486X%28196909%292%3A90%3A2%3C372%3AUMIF%3E2.0.CO%3B2-C [5] http://links.jstor.org/sici?sici=0003-486X%28193601%292%3A37%3Al%3C116%3AOROO%3E2.0.CO%3B2-Y [6] http://links.jstor.org/sici?sici=0002-9947%28193703%2941%3A2%3C208%3AOROOI%3E2.0.CO%3B2-9 [7] http://links.jstor.org/sici?sici=0003-486X%28194310%292%3A44%3A4%3C716%3AOROOI%3E2.0.CO%3B2-O [8] http://links.jstor.org/sici?sici=0003-486X%28196707%292%3A86%3Al%3C138%3AROUHAA%3E2.0.CO%3B2-6 [9] http://eom.springer.de/V/v096900.htm [10] http://links.jstor.org/sici?sici=0003-486X%28193601%292%3A37%3Al%3Clll%3AOACTFR%3E2.0.CO%3B2-S [II] http://www.numdam.org/item?id=CM_1939 6 1_0 [12] http://links.jstor.org/sici?sici=0003-486X%28194001%292%3A41%3Al%3C94%3AOROOI%3E2.0.CO%3B2-U [13] http://links.jstor.org/sici?sici=0003-486X%28194310%292%3A44%3A4%3C709%3AOSAPOO%3E2.0.CO%3B2-L [14] http://links.jstor.org/sici?sici=0003-486X%28194904%292%3A50%3A2%3C401%3AOROORT%3E2.0.CO%3B2-H [15] http://iml.univ-mrs.fr/~wasserm/OHS.ps C*-algebra 150 C* -algebra C*-algebras (pronounced "C-star") are an important area of research in functional analysis, a branch of mathematics. The prototypical example of a C*-algebra is a complex algebra A of linear operators on a complex Hilbert space with two additional properties: • A is a topologically closed set in the norm topology of operators. • A is closed under the operation of taking adjoints of operators. It is generally believed that C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables. This line of research began with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently John von Neumann attempted to establish a general framework for these algebras which culminated in a series of papers on rings of operators. These papers considered a special class of C*-algebras which are now known as von Neumann algebras. Around 1943, the work of Israel Gelfand and Mark Naimark yielded an abstract characterisation of C*-algebras making no reference to operators. C*-algebras are now an important tool in the theory of unitary representations of locally compact groups, and are also used in algebraic formulations of quantum mechanics. Abstract characterization We begin with the abstract characterization of C*-algebras given in the 1943 paper by Gelfand and Naimark. A C*-algebra, A, is a Banach algebra over the field of complex numbers, together with a map, * : A — > A, called an involution. The image of an element x of A under the involution is written x*. Involution has the following properties: • For all x,y in A: (x + y)* = x*+y* (xy)* = y*x* • For every X in C and every xinA: (Arc)* = Ax*. • For all x in A (x*y = x • The C*— identity holds for all x in A: ||:r*:r|| = ||zr||||x*||. Note that the C* identity is equivalent to: for all x in A: ||:c:r*|| = ||zr||||x*||. This relation is equivalent to ||xa;*|| = ||x|| 2 > which is sometimes called the B*-identity. For history behind the names C*- and B*-algebras, see the history section below. The C*-identity is a very strong requirement. For instance, together with the spectral radius formula, it implies the C*-norm is uniquely determined by the algebraic structure: ||x|| = ||x*x|| = sup{|A| : x*x — A 1 is not invertible}. A bounded linear map, jt : A — > B, between C*-algebras A and B is called a *-homomorphism if • For x and y in A ■K{xy) = 7r(x)Tr(y) C*-algebra 151 • For x in A TT(X*) = 7r(x)* In the case of C*-algebras, any *-homomorphism n between C*-algebras is non-expansive, i.e. bounded with norm < 1. Furthermore, an injective *-homomorphism between C*-algebras is isometric. These are consequences of the C*-identity. A bijective *-homomorphism n is called a C*-isomorphism, in which case A and B are said to be isomorphic. Some history: B* -algebras and C* -algebras The term B*-algebra was introduced by C. E. Rickart in 1946 to describe Banach *-algebras that satisfy the condition 2 • I be x*ll = llxll for all x in the given B*-algebra. (B* -condition) This condition automatically implies that the *-involution is isometric, that is, llx*ll = llxll. Hence llx x*ll = llxll llx*ll, and therefore, a B*-algebra is a C*-algebra. Conversely, the C*-condition implies the B*-condition. This is nontrivial, and can be proved without using the condition llxll=llx*ll. (For details, see R. S. Doran, V. A. Belfi, Characterizations of 'C*- Algebras --- the Gelfand-Naimark Theorems, CRC, 1986.) For these reasons, the term B*-algebra is rarely used in current terminology, and has been replaced by the term 'C* algebra'. The term C*-algebra was introduced by I. E. Segal in 1947 to describe norm-closed subalgebras of B(H), namely, the space of bounded operators on some Hilbert space H. 'C stood for 'closed'. Examples Finite-dimensional C* -algebras The algebra M (C) of n-by-n matrices over C becomes a C*-algebra if we consider matrices as operators on the Euclidean space, C' , and use the operator norm 11.11 on matrices. The involution is given by the conjugate transpose. More generally, one can consider finite direct sums of matrix algebras. In fact, all finite dimensional C*-algebras are of this form. The self-adjoint requirement means finite-dimensional C*-algebras are semisimple, from which fact one can deduce the following theorem of Artin— Wedderburn type: Theorem. A finite-dimensional C*-algebra, A, is canonically isomorphic to a finite direct sum A= © Ae e6min A where min A is the set of minimal nonzero self-adjoint central projections of A. Each C*-algebra, Ae, is isomorphic (in a noncanonical way) to the full matrix algebra M (C). The finite family indexed on min A given by {dim(e)} is called the dimension vector of A. This vector uniquely determines the isomorphism class of a finite-dimensional C*-algebra. In the language of K-theory, this vector is the positive cone of the K n group of A. C*-algebra 152 C*-algebras of operators The prototypical example of a C*-algebra is the algebra B(H) of bounded (equivalently continuous) linear operators defined on a complex Hilbert space H; here x* denotes the adjoint operator of the operator x : H — > H. In fact, every C*-algebra, A, is *-isomorphic to a norm-closed adjoint closed subalgebra of B(H) for a suitable Hilbert space, H; this is the content of the Gelfand— Naimark theorem. Commutative C*-algebras Let X be a locally compact Hausdorff space. The space C (X) of complex-valued continuous functions on X that vanish at infinity (defined in the article on local compactness) form a commutative C*-algebra CJX) under pointwise multiplication and addition. The involution is pointwise conjugation. CJX) has a multiplicative unit element if and only if X is compact. As does any C*-algebra, CJX) has an approximate identity. In the case of CJX) this is immediate: consider the directed set of compact subsets of X, and for each compact K let /be a function of compact support which is identically 1 on K. Such functions exist by the Tietze extension theorem which applies to locally compact Hausdorff spaces, {f } is an approximate identity. K K The Gelfand representation states that every commutative C*-algebra is *-isomorphic to the algebra CJX), where X is the space of characters equipped with the weak* topology. Furthermore if CJX) is isomorphic to CJY) as C*-algebras, it follows that X and Y are homeomorphic. This characterization is one of the motivations for the noncommutative topology and noncommutative geometry programs. C*-algebras of compact operators Let H be a separable infinite-dimensional Hilbert space. The algebra K(H) of compact operators on H is a norm closed subalgebra of B(H). It is also closed under involution; hence it is a C*-algebra. Concrete C*-algebras of compact operators admit a characterization similar to Wedderburn's theorem for finite dimensional C*-algebras. Theorem. If A is a C*-subalgebra of K(H), then there exists Hilbert spaces {H.} . such that A is isomorphic to the following direct sum 16/ where the (C*-)direct sum consists of elements (T.) of the Cartesian product n K{H) with II7\II — > 0. Though K(H) does not have an identity element, a sequential approximate identity for K(H) can be easily displayed. J H = e 2 . ;h natun k > n 2 To be specific, H is isomorphic to the space of square summable sequences / ; we may assume that 2 For each natural number n let H be the subspace of sequences of / which vanish for indices and let be the orthogonal projection onto H . The sequence {e } is an approximate identity for K(H). K(H) is a two-sided closed ideal of B(H). For separable Hilbert spaces, it is the unique ideal. The quotient of B(H) by K(H) is the Calkin algebra. C*-algebra 153 C*-enveloping algebra Given a B*-algebra A with an approximate identity, there is a unique (up to C*-isomorphism) C*-algebra E(A) and *-morphism n from A into E(A) which is universal, that is, every other B*-morphism n ' : A — > B factors uniquely through jt. The algebra E(A) is called the C*-enveloping algebra of the B*-algebra A. Of particular importance is the C*-algebra of a locally compact group G. This is defined as the enveloping C*-algebra of the group algebra of G. The C*-algebra of G provides context for general harmonic analysis of G in the case G is non-abelian. In particular, the dual of a locally compact group is defined to be the primitive ideal space of the group C*-algebra. See spectrum of a C*-algebra. von Neumann algebras von Neumann algebras, known as W* algebras before the 1960s, are a special kind of C*-algebra. They are required to be closed in the weak operator topology, which is weaker than the norm topology. Their study is a specialized area of functional analysis. Properties of C*-algebras C*-algebras have a large number of properties that are technically convenient. These properties can be established by use the continuous functional calculus or by reduction to commutative C*-algebras. In the latter case, we can use the fact that the structure of these is completely determined by the Gelfand isomorphism. • The set of elements of a C*-algebra A of the form x*x forms a closed convex cone. This cone is identical to the elements of the form x x*. Elements of this cone are called non-negative (or sometimes positive, even though this terminology conflicts with its use for elements of R.) • The set of self-adjoint elements of a C*-algebra A naturally has the structure of a partially ordered vector space; the ordering is usually denoted >. In this ordering, a self-adjoint element x of A satisfies x > if and only if the spectrum of x is non-negative. Two self-adjoint elements x and y of A satisfy x>^ifx-j>0. • Any C*-algebra A has an approximate identity. In fact, there is a directed family {e } of self-adjoint elements A. A. fc I of A such that X&\ — > X < e^ < e^ < 1 whenever A < fi. In case A is separable, A has a sequential approximate identity. More generally, A will have a sequential approximate identity if and only if A contains a strictly positive element, i.e. a positive element h such that hAh is dense in A. • Using approximate identities, one can show that the algebraic quotient of a C*-algebra by a closed proper two-sided ideal, with the natural norm, is a C*-algebra. • Similarly, a closed two-sided ideal of a C*-algebra is itself a C*-algebra. C*-algebra 154 Type for C*-algebras A C*-algebra A is of type I if and only if for all non-degenerate representations x of A the von Neumann algebra jt(A)" (that is, the bicommutant of Jt(A)) is a type I von Neumann algebra. In fact it is sufficient to consider only factor representations, i.e. representations n for which jt(A)" is a factor. A locally compact group is said to be of type I if and only if its group C*-algebra is type I. However, if a C*-algebra has non-type I representations, then by results of James Glimm it also has representations of type II and type III. Thus for C*-algebras and locally compact groups, it is only meaningful to speak of type I and non type I properties. C*-algebras and quantum field theory In quantum field theory, one typically describes a physical system with a C*-algebra A with unit element; the self-adjoint elements of A (elements x with x* = x) are thought of as the observables, the measurable quantities, of the system. A state of the system is defined as a positive functional on A (a C-linear map cp : A — > C with q>(u* u) > for all m€A) such that cp(l) = 1. The expected value of the observable x, if the system is in state q>, is then cp(x). See Local quantum physics. References • W. Arveson, An Invitation to C*-Algebra, Springer- Verlag, 1976. ISBN 0-387-90176-0. An excellent introduction to the subject, accessible for those with a knowledge of basic functional analysis. • A. Connes, Non-commutative geometry (http://www.alainconnes.org/docs/book94bigpdf.pdf), ISBN 0-12-185860-X. This book is widely regarded as a source of new research material, providing much supporting intuition, but it is difficult. • J. Dixmier, Les C*-algebres et leurs representations, Gauthier-Villars, 1969. ISBN 0-7204-0762-1. This is a somewhat dated reference, but is still considered as a high-quality technical exposition. It is available in English from North Holland press. • G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, 1972. ISBN 0-471-23900-3. Mathematically rigorous reference which provides extensive physics background. • A.I. Shtern (2001), "C* algebra" (http://eom.springer.de/c/c020020.htm), in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104 • S. Sakai, C*-algebras and W*-algebras , Springer (1971) ISBN 3-540-63633-1 Quasi-Hopf algebra 155 Quasi-Hopf algebra A quasi-Hopf algebra is a generalization of a Hopf algebra, which was defined by the Russian mathematician Vladimir Drinfeld in 1989. A quasi-Hopf algebra is a quasi-bialgebra Bj^ = (J\_^ A, £, c&)for which there exist a y f3 £ ^4 and a bijective antihomomorphism S (antipode) of J^ such that ^ S(bi)aci = £(a)a i Y^biPSia) = e{a)j3 i for all a £ j\ an d where A(a) = J2bi®Ci i and i Y / S(P j )aQ j pS(R j )=I. 3 where the expansions for the quantities ^>and qj— lare given by i and <S>- 1 =Y J P3®Qj® R 3- 3 As for a quasi-bialgebra, the property of being quasi-Hopf is preserved under twisting. Usage Quasi-Hopf algebras form the basis of the study of Drinfeld twists and the representations in terms of F-matrices associated with finite-dimensional irreducible representations of quantum affine algebra. F-matrices can be used to factorize the corresponding R-matrix. This leads to applications in Statistical mechanics, as quantum affine algebras, and their representations give rise to solutions of the Yang-Baxter equation, a solvability condition for various statistical models, allowing characteristics of the model to be deduced from its corresponding quantum affine algebra. The study of F-matrices has been applied to models such as the Heisenberg XXZ model in the framework of the algebraic Bethe ansatz. It provides a framework for solving two-dimensional integrable models by using the Quantum inverse scattering method. References • Vladimir Drinfeld, Quasi-Hopf algebras, Leningrad Math J. 1 (1989), 1419-1457 • J.M. Maillet and J. Sanchez de Santos, Drinfeld Twists and Algebraic Bethe Ansatz, Amer. Math. Soc. Transl. (2) Vol. 201, 2000 Quasitriangular Hopf algebra 156 Quasitriangular Hopf algebra In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of J{ (g //"such that • R A(x) = (T o A)(x) R for all x £ H . where ^is the coproduct on //, and the linear map T : H ®H -> # ® # is given by T{x ® y) = y®zr, • (A®l)(i?)= J R 13 i?23' • (l®A)(i?)= J R 13 i?i2» where i? 12 = <j) 12 (R), R 13 = <j>i 3 (R), and i? 23 = <j) 23 (R), where (j) l2 : H ® H ^ H ® H ® H , 013 : H®H^H®H®H, and 2 3 '-H®H—>H®H®H,zre algebra morphisms determined by 0i 2 (a® b) = ■ a ® 6 ® 1, 013 (a® b) = a ® 1 ® 6, 023 («® &) = 1 ® a ® 6. flis called the R-matrix. As a consequence of the properties of quasitriangularity, the R-matrix, R, is a solution of the Yang-Baxter equation (and so a module V of H can be used to determine quasi-invariants of braids, knots and links). Also as a consequence of the properties of quasitriangularity, (e ® l)i? = (1 ® ejR = 1 £ H ; moreover i? _1 = (5 ® 1)(-R)> i? = (1 (g> S')( J R _1 ), and (£ ® jS^fR) = i? . One may further show that the antipode S must be a linear isomorphism, and thus S A 2 is an automorphism. In fact, S A 2 is given by conjugating by an invertible element: S(x) = UXU _1 where u = m(S ® l)i? 21 (cf. Ribbon Hopf algebras). It is possible to construct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfel'd quantum double construction. Twisting The property of being a quasi-triangular Hopf algebra is preserved by twisting via an invertible element ■^ / j J W /« fc ^ w /I suc h ma t l £ (g id\p = (id g) e)_F = land satisfying the cocycle condition i (F ® 1) o (A ® id)F = (1 ® F) o (id ® A)F Furthermore, u = ,2-* / ^{ji) is invertible and the twisted antipode is given by S'(a) = uS(a)u~ 1 , with the i twisted comultiplication, R-matrix and co-unit change according to those defined for the quasi-triangular Quasi-Hopf algebra. Such a twist is known as an admissible (or Drinfel'd) twist. Notes [1] Montgomery & Schneider (2002), p. 72 (http://books.google.com/books?id=I3IK9U5Co_0C&pg=PA72&dq="Quasitriangular"). References • Susan Montgomery, Hans-Jurgen Schneider. New directions in Hopf algebras, Volume 43. Cambridge University Press, 2002. ISBN 9780521815123 Ribbon Hopf algebra 157 Ribbon Hopf algebra A ribbon Hopf algebra (A, m, A, U, £, S, 1Z, i^)is a quasitriangular Hopf algebra which possess an invertible central element v more commonly known as the ribbon element, such that the following conditions hold: v = uS(u), S(v) = l>, e{y) = 1 A(v) = (R 21 n 12 )- 1 (v®v) where u = m(S ® id)(7^2i)- Note that the element u exists for any quasitriangular Hopf algebra, and uS(u) must always be central and satisfies S(uS(u)) = uS(u),e(uS(u)) = l,A(uS(u)) = (Tl 21 Tl 12 )- 2 (uS(u) ® uS(u)), so that all that is required is that it have a central square root with the above properties. Here A is a vector space m is the multiplication map rn \ A® A — > A A is the co-product map A : A — > A <S> A U is the unit operator u : C — > A E is the co-unit operator £ \ A — > C S is the antipode S : A — > A 72. is a universal R matrix We assume that the underlying field J{ is (£ See also • Quasitriangular Hopf algebra • Quasi-triangular Quasi-Hopf algebra References • Altschuler, D., Coste, A.: Quasi-quantum groups, knots, three-manifolds and topological field theory. Commun. Math. Phys. 150 1992 83-107 http://arxiv.org/pdf/hep-th/9202047 • Chari, V.C., Pressley, A.: A Guide to Quantum Groups Cambridge University Press, 1994 ISBN 0-521-55884-0. • Vladimir Drinfeld, Quasi-Hopf algebras , Leningrad Math J. 1 (1989), 1419-1457 • Shahn Majid : Foundations of Quantum Group Theory Cambridge University Press, 1995 Quasi-triangular Quasi-Hopf algebra 158 Quasi-triangular Quasi-Hopf algebra A quasi-triangular quasi-Hopf algebra is a specialized form of a quasi-Hopf algebra defined by the Ukrainian mathematician Vladimir Drinfeld in 1989. It is also a generalized form of a quasi-triangular Hopf algebra. A quasi-triangular quasi-Hopf algebra is a set "hij^ = (A, i?, A, e, $) where B^ = (A, A, £, 3>)is a quasi-Hopf algebra and J? £ A gj ,4. known as the R-matrix, is an invertible element such that RA(a) = o- o A(a)R, a€A a : ,4®.4^.4® A x ®y — > ?/ ® x so that cr is the switch map and (A®id)R = $321^13^^32^23^123 (id ® A)i? = $ 2 - 3 1 1J Rl3$213^12$r23 where $ abc = x Q ® X b ® X c and $ 123 = $ = Xi®X 2 ®X 3 e^l®^l®^l. The quasi-Hopf algebra becomes triangular if in addition, i?2ii?i2 = 1- The twisting of Ti^ by F (H A® A* 5 the same as for a quasi-Hopf algebra, with the additional definition of the twisted 7?-matrix A quasi-triangular (resp. triangular) quasi-Hopf algebra with Q = lis a quasi-triangular (resp. triangular) Hopf algebra as the latter two conditions in the definition reduce the conditions of quasi-triangularity of a Hopf algebra . Similarly to the twisting properties of the quasi-Hopf algebra, the property of being quasi-triangular or triangular quasi-Hopf algebra is preserved by twisting. See also • Quasitriangular Hopf algebra • Ribbon Hopf algebra References • Vladimir Drinfeld, Quasi-Hopf algebras, Leningrad Math J. 1 (1989), 1419-1457 • J.M. Maillet and J. Sanchez de Santos, Drinfeld Twists and Algebraic Bethe Ansatz, Amer. Math. Soc. Transl. (2) Vol. 201, 2000 Quantum inverse scattering method 159 Quantum inverse scattering method Quantum inverse scattering method relates two different approaches: l)Inverse scattering transform is a method of solving classical integrable differential equations of evolutionary type. Important concept is Lax representation. 2) Bethe ansatz is a method of solving quantum models in one space and one time dimension. Quantum inverse scattering method starts by quantization of Lax representation and reproduce results of Bethe ansatz. Actually it permits to rewrite Bethe ansatz in a new form: algebraic Bethe ansatz. This led to further progress in understanding of Heisenberg model (quantum), quantum Nonlinear Schrodinger equation (also known as Bose gas with delta interaction) and Hubbard model. Theory of correlation functions was developed. In mathematics in led to formulation of quantum groups. Especially interesting one is Yangian. In mathematics, the quantum inverse scattering method is a method for solving integrable models in 1+1 dimensions introduced by L. D. Faddeev in about 1979. References • Faddeev, L. (1995), "Instructive history of the quantum inverse scattering method" , Acta Applicandae Mathematicae. 39 (1): 69-84, MR1329554, ISSN 0167-8019 • Korepin, V. E.; Bogoliubov, N. M.; Izergin, A. G. (1993), Quantum inverse scattering method and correlation ni functions , Cambridge Monographs on Mathematical Physics, Cambridge University Press, MR1245942, ISBN 978-0-521-37320-3 References [1] http://dx.doi.org/10.1007/BF00994626 [2] http://www. Cambridge. org/catalogue/catalogue.asp?isbn=9780521586467 Grassmann algebra 160 Grassmann algebra In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. Like the cross product, and the scalar triple product, the exterior product of vectors is used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs. Also, like the cross product, the exterior product is alternating, meaning that u a u = for all vectors u, or equivalently u a v = -v a u for all vectors u and v. In linear algebra, the exterior product provides an abstract algebraic manner for describing the determinant and the minors of a linear transformation that is basis-independent, and is fundamentally related to ideas of rank and linear independence. The exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann ) of a given vector space V over a field K is the unital associative algebra A(V) generated by the exterior product. It is widely used in contemporary geometry, especially differential geometry and algebraic geometry through the algebra of differential forms, as well as in multilinear algebra and related fields. In terms of category theory, the exterior algebra is a type of functor on vector spaces, given by a universal construction. The universal construction allows the exterior algebra to be defined, not just for vector spaces over a field, but also for modules over a commutative ring, and for other structures of interest. The exterior algebra is one example of a bialgebra, meaning that its dual space also possesses a product, and this dual product is compatible with the wedge product. This dual algebra is precisely the algebra of alternating multilinear forms on V, and the pairing between the exterior algebra and its dual is given by the interior product. Motivating examples Areas in the plane 2 The Cartesian plane R is a vector space equipped with a basis consisting of a pair of unit vectors The area of a parallelogram in terms of the determinant of the matrix of coordinates of two of its vertices. ei = (l,0), e 2 = (0,l). Suppose that v — v 1 e 1 +v 2 e 2 , w — w 1 e 1 +w 2 e 2 2 axe a pair of given vectors in R , written in components. There is a unique parallelogram having v and w as two of its sides. The area of this parallelogram is given by the standard determinant formula: Grassmann algebra 161 A = det [v wl = \v1W2 — V2W1 \. Consider now the exterior product of v and w: v A w = (v^j + v 2 e 2 ) A (w 1 e 1 + w 2 e 2 ) = vitviei A ei + v\w 2 Gi A e 2 + v 2 w\e 2 A ei + v 2 w 2 e 2 A e 2 — (v 1 w 2 - u 2 ^i)ei A e 2 where the first step uses the distributive law for the wedge product, and the last uses the fact that the wedge product is alternating, and in particular e a e = -e a e . Note that the coefficient in this last expression is precisely the determinant of the matrix [v w]. The fact that this may be positive or negative has the intuitive meaning that v and w may be oriented in a counterclockwise or clockwise sense as the vertices of the parallelogram they define. Such an area is called the signed area of the parallelogram: the absolute value of the signed area is the ordinary area, and the sign determines its orientation. The fact that this coefficient is the signed area is not an accident. In fact, it is relatively easy to see that the exterior product should be related to the signed area if one tries to axiomatize this area as an algebraic construct. In detail, if A(v,w) denotes the signed area of the parallelogram determined by the pair of vectors v and w, then A must satisfy the following properties: 1 . A(a\,bw) = a b A(v,w) for any real numbers a and b, since rescaling either of the sides rescales the area by the same amount (and reversing the direction of one of the sides reverses the orientation of the parallelogram). 2. A(v,v) = 0, since the area of the degenerate parallelogram determined by v (i.e., a line segment) is zero. 3. A(w,v) = -A(v,w), since interchanging the roles of v and w reverses the orientation of the parallelogram. 4. A(v + aw,w) = A(v,w), since adding a multiple of w to v affects neither the base nor the height of the parallelogram and consequently preserves its area. 5. A(e , e ) - 1, since the area of the unit square is one. With the exception of the last property, the wedge product satisfies the same formal properties as the area. In a certain sense, the wedge product generalizes the final property by allowing the area of a parallelogram to be compared to that of any "standard" chosen parallelogram. In other words, the exterior product in two-dimensions is a T31 basis-independent formulation of area. Cross and triple products For vectors in R , the exterior algebra is closely related to the cross product and triple product. Using the standard basis {e , e , e }, the wedge product of a pair of vectors u = uiei + u 2 e 2 + u 3 e 3 and is v = viei + v 2 e 2 + v 3 e 3 u A v = (u 1 v 2 - u 2 v 1 )(e 1 /\e 2 ) + (u 3 v 1 -u 1 v 3 )(e 3 Ae 1 ) + (u 2 v 3 - u 3 v 2 )(e 2 Ae 3 ) 2 3 where {e A e , e A e , e A e } is the basis for the three-dimensional space A (R ). This imitates the usual definition of the cross product of vectors in three dimensions. Bringing in a third vector w = wiei + w 2 e 2 + w 3 e 3 , the wedge product of three vectors is uAvAw = (u 1 v 2 w 3 +u 2 v 3 w 1 +u 3 v 1 w 2 -u 1 v 3 w 2 -u 2 v 1 w 3 -u 3 v 2 w 1 )(e 1 Ae 2 Ae 3 ) 3 3 where e A e A e is the basis vector for the one-dimensional space A (R ). This imitates the usual definition of the triple product. Grassmann algebra 162 The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations. The cross product uxv can be interpreted as a vector which is perpendicular to both u and v and whose magnitude is equal to the area of the parallelogram determined by the two vectors. It can also be interpreted as the vector consisting of the minors of the matrix with columns u and v. The triple product of u, v, and w is geometrically a (signed) volume. Algebraically, it is the determinant of the matrix with columns u, v, and w. The exterior product in three-dimensions allows for similar interpretations. In fact, in the presence of a positively oriented orthonormal basis, the exterior product generalizes these notions to higher dimensions. Formal definitions and algebraic properties The exterior algebra A(V) over a vector space V is defined as the quotient algebra of the tensor algebra by the two-sided ideal / generated by all elements of the form x ® x such that x € V. Symbolically, \(V) := T(V)/I. The wedge product a of two elements of A(V) is defined by a A (3 — a® j3 (modi). Anticommutativity of the wedge product The wedge product is alternating on elements of V, which means that x a x = for all x G V. It follows that the product is also anticommutative on elements of V, for supposing that x, y € V, = (x + y)A(x + y) = xAx + xAy + yAx + yAy = xAy + yAx whence x Ay — —y A x. More generally, if x , x , ..., x are elements of V, and o is a permutation of the integers [l,...,fe], then z CT (i) A x^) A ■ ■ ■ A x a ( k ) = sgn(o-)a;i A x 2 A ■ ■ ■ A x k , where sgn(o) is the signature of the permutation o. The exterior power The kth exterior power of V, denoted A (V), is the vector subspace of A(V) spanned by elements of the form Xi A X2 A . . . A xjc, Xi G V, i = 1, 2, . . . , k. If a G A (V), then a is said to be a fe-multivector. If, furthermore, a can be expressed as a wedge product of k elements of V, then a is said to be decomposable. Although decomposable multivectors span A (V), not every k 4 element of A (V) is decomposable. For example, in R , the following 2-multivector is not decomposable: a = e\ A €2 + es A e^ . (This is in fact a symplectic form, since a a a * 0. ) Grassmann algebra 163 Basis and dimension If the dimension of Vis n and {<? ,...,e } is a basis of V, then the set 1 n {e h A e i2 A - ■ ■ A e ik | 1 < i x < i 2 < ■ ■ ■ < i k < n} is a basis for A (V). The reason is the following: given any wedge product of the form V\ A ■ ■ ■ A Vk then every vector v . can be written as a linear combination of the basis vectors e .; using the bilinearity of the wedge J ' product, this can be expanded to a linear combination of wedge products of those basis vectors. Any wedge product in which the same basis vector appears more than once is zero; any wedge product in which the basis vectors do not appear in the proper order can be reordered, changing the sign whenever two basis vectors change places. In general, the resulting coefficients of the basis fe-vectors can be computed as the minors of the matrix that describes the vectors v. in terms of the basis e.. } i k k By counting the basis elements, the dimension of A (V) is the binomial coefficient C(n,k). In particular, A (V) = {0} for k > n. Any element of the exterior algebra can be written as a sum of multivectors. Hence, as a vector space the exterior algebra is a direct sum A(v) = a°(v) e a\v) e a 2 (v) e ■ ■ ■ e A n (v) e by convention / coefficients, which is 2" (where by convention A (V) = K and A (V) = V), and therefore its dimension is equal to the sum of the binomial Rank of a multivector If a G A (V), then it is possible to express a as a linear combination of decomposable multivectors: where each a is decomposable, say a® =a? ) A---Aa$, 1 = 1,2,..., a. The rank of the multivector a is the minimal number of decomposable multivectors in such an expansion of a. This is similar to the notion of tensor rank. Rank is particularly important in the study of 2-multivectors (Sternberg 1974, §111.6) (Bryant et al. 1991). The rank of a 2-multivector a can be identified with half the rank of the matrix of coefficients of a in a basis. Thus if e. is a i basis for V, then a can be expressed uniquely as a = ^2 a v e i A e J where a..= -a (the matrix of coefficients is skew-symmetric). The rank of the matrix a is therefore even, and is 'J J' V twice the rank of the form a. In characteristic 0, the 2-multivector a has rank p if and only if a A ■ ■ ■ A a ^ 0. Grassmann algebra 164 Graded structure The wedge product of a fe-multivector with a p-multivector is a (£+/?)-multivector, once again invoking bilinearity. As a consequence, the direct sum decomposition of the preceding section A(V) = A°(V) © A\V) © A 2 (V) © ■ ■ ■ © A n (V) gives the exterior algebra the additional structure of a graded algebra. Symbolically, (A fe (V)) A (A P (V)) C A k+p (V). Moreover, the wedge product is graded anticommutative, meaning that if a G A (V) and p G A P (V), then ahf3 = (-l) kp (3Aa. In addition to studying the graded structure on the exterior algebra, Bourbaki (1989) studies additional graded structures on exterior algebras, such as those on the exterior algebra of a graded module (a module that already carries its own gradation). Universal property Let V be a vector space over the field K. Informally, multiplication in A(V) is performed by manipulating symbols and imposing a distributive law, an associative law, and using the identity v a v = for v € V. Formally, A(V) is the "most general" algebra in which these rules hold for the multiplication, in the sense that any unital associative ^-algebra containing V with alternating multiplication on V must contain a homomorphic image of A(V). In other words, the exterior algebra has the following universal property: Given any unital associative ^-algebra A and any ^-linear map j : V — > A such that j(v)j(v) = for every v in V, then there exists precisely one unital algebra homomorphism/: A(V) — > A such that j(v) =f(i(v)) for all v in V. V ( -~A(V) j i :/ > + A To construct the most general algebra that contains V and whose multiplication is alternating on V, it is natural to start with the most general algebra that contains V, the tensor algebra T{V), and then enforce the alternating property by taking a suitable quotient. We thus take the two-sided ideal / in T(V) generated by all elements of the form v<E>v for v in V, and define A(V) as the quotient A(V) = T(V)/I (and use A as the symbol for multiplication in A(V)). It is then straightforward to show that A(V) contains V and satisfies the above universal property. As a consequence of this construction, the operation of assigning to a vector space V its exterior algebra A(V) is a functor from the category of vector spaces to the category of algebras. Rather than defining A(V) first and then identifying the exterior powers A (V) as certain subspaces, one may alternatively define the spaces A (V) first and then combine them to form the algebra A(V). This approach is often used in differential geometry and is described in the next section. Grassmann algebra 165 Generalizations Given a commutative ring R and an 7?-module M, we can define the exterior algebra A(M) just as above, as a suitable quotient of the tensor algebra T(M). It will satisfy the analogous universal property. Many of the properties of A(M) also require that M be a projective module. Where finite-dimensionality is used, the properties further require that M be finitely generated and projective. Generalizations to the most common situations can be found in (Bourbaki 1989). Exterior algebras of vector bundles are frequently considered in geometry and topology. There are no essential differences between the algebraic properties of the exterior algebra of finite-dimensional vector bundles and those of the exterior algebra of finitely-generated projective modules, by the Serre-Swan theorem. More general exterior algebras can be defined for sheaves of modules. Duality Alternating operators Given two vector spaces V and X, an alternating operator (or anti-symmetric operator) from v to X is a multilinear map / : V k -> X such that whenever v ,...,v. are linearly dependent vectors in V, then f(v 1 ,...,v k ) = A well-known example is the determinant, an alternating operator from (K ) to K. The map w : V k -> A k (V) which associates to k vectors from V their wedge product, i.e. their corresponding fe-vector, is also alternating. In fact, this map is the "most general" alternating operator defined on V: given any other alternating operator/: v — > X, there exists a unique linear map cp: A (V) — » X with / = cp o w. This universal property characterizes the space A (V) and can serve as its definition. Alternating multilinear forms The above discussion specializes to the case when X = K, the base field. In this case an alternating multilinear function / : V k -> K is called an alternating multilinear form. The set of all alternating multilinear forms is a vector space, as the sum of two such maps, or the product of such a map with a scalar, is again alternating. By the universal property of the exterior power, the space of alternating forms of degree k on V is naturally isomorphic with the dual vector space (A V) . If V is finite-dimensional, then the latter is naturally isomorphic to A (V ). In particular, the dimension of the space of anti-symmetric maps from 'v to K is the binomial coefficient n choose k. Under this identification, the wedge product takes a concrete form: it produces a new anti-symmetric map from two given ones. Suppose co : v —> K and r) : V 1 — > K are two anti-symmetric maps. As in the case of tensor products of multilinear maps, the number of variables of their wedge product is the sum of the numbers of their variables. It is defined as follows: u A V = ^n — r-Alt(ci; <g> 17) k\ m\ where the alternation Alt of a multilinear map is defined to be the signed average of the values over all the permutations of its variables: Grassmann algebra 166 A\t(uj)(x 1 , . . . , x k ) = — ^^ sgn(a) ^(x^i), . . . , x CT(fe) ). k\ This definition of the wedge product is well-defined even if the field K has finite characteristic, if one considers an equivalent version of the above that does not use factorials or any constants: a;A7j(xi, . . .,x k+m ) = ^ sgn((7)w(i ff( i), . . .,x<T( k ))r)(x a ( k+1 ), . . .,x^ k+m )), where here Sh C S is the subset of (k,m) shuffles: permutations o of the set { l,2,...,k+m] such that o(l) < o(2) K,f7l K~rf7l ron < ... < o(lc), and a(k+l) < o(k+2)< ... <a(k+m). Bialgebra structure In formal terms, there is a correspondence between the graded dual of the graded algebra A(V) and alternating multilinear forms on V. The wedge product of multilinear forms defined above is dual to a coproduct defined on A(V), giving the structure of a coalgebra. The coproduct is a linear function A : A(V) — > A(V) <E> A(V) given on decomposable elements by k A(X 1 A---Ax k ) = ^ ^ S g n {< 7 )( X <r(l) A --- Ax <r(j>))®( X r(p+l) A --- Ax c7(k))- p—0 aEShp^k-p For example, A(xi) = l®x 1 +x 1 ® 1, A(xi A x 2 ) = 1 ® (i] A x 2 ) + xi ® x 2 — x 2 ® xi + (xi A x 2 ) ® 1. This extends by linearity to an operation defined on the whole exterior algebra. In terms of the coproduct, the wedge product on the dual space is just the graded dual of the coproduct: (a A /3)(i! A . . . A x k ) = (a ® /3) (A(xi A ... A x k )) where the tensor product on the right-hand side is of multilinear linear maps (extended by zero on elements of incompatible homogeneous degree: more precisely, aAp = e o (a<E>|3) o A, where e is the counit, as defined presently). The counit is the homomorphism e : A(V) — > K which returns the 0-graded component of its argument. The coproduct and counit, along with the wedge product, define the structure of a bialgebra on the exterior algebra. With an antipode defined on homogeneous elements by S(x) = (-1) egx x, the exterior algebra is furthermore a Hopf algebra. Interior product Suppose that V is finite-dimensional. If V* denotes the dual space to the vector space V, then for each a € V , it is possible to define an antiderivation on the algebra A(V), i a : A k V -> A fc -V. This derivation is called the interior product with a, or sometimes the insertion operator, or contraction by a. k * Suppose that w € A V. Then w is a multilinear mapping of V to K, so it is defined by its values on the fc-fold Cartesian product V x V x ... x V . If m, , «„, ..., u, , are k-1 elements of V , then define 1 2 k-l (i Q w)(ui, u 2 . . . , ii fe _i) = w(a, tii, "2, ■ ■ ■ , Uk-i)- Additionally, let i f= whenever/is a pure scalar (i.e., belonging to A V). Grassmann algebra 167 Axiomatic characterization and properties The interior product satisfies the following properties: 1 . For each k and each a G V , i a : A k V -► A k ~ 1 V. (By convention, A - = 0.) 1 * 2. If v is an element of V ( = A V), then / v = a(v) is the dual pairing between elements of V and elements of V . * a 3. For each a GV,i is a graded derivation of degree -1: a i a (a A 6) = (« Q a) A & + (-l) dega a A (i a b). In fact, these three properties are sufficient to characterize the interior product as well as define it in the general infinite-dimensional case. Further properties of the interior product include: * i a ° ia = 0. * i a °*/3 = -i/3 °i a - Hodge duality Suppose that V has finite dimension n. Then the interior product induces a canonical isomorphism of vector spaces A k (V*) ® A n (V) -> A"-*(V). In the geometrical setting, a non-zero element of the top exterior power A (V) (which is a one-dimensional vector space) is sometimes called a volume form (or orientation form, although this term may sometimes lead to ambiguity.) Relative to a given volume form o, the isomorphism is given explicitly by a G A. h (V*) ^ i a tr G A n_ *(V). If, in addition to a volume form, the vector space V is equipped with an inner product identifying V with V , then the resulting isomorphism is called the Hodge dual (or more commonly the Hodge star operator) * : A k (V) -» A n ~ k (V). k k The composite of * with itself maps A (V) — > A (V) and is always a scalar multiple of the identity map. In most applications, the volume form is compatible with the inner product in the sense that it is a wedge product of an orthonormal basis of V. In this case, * o * : A k (V) -> A k (V) = (_l)*("-*)+9/ where / is the identity, and the inner product has metric signature (p,q) — p plusses and q minuses. Inner product For V a finite-dimensional space, an inner product on V defines an isomorphism of V with V , and so also an isomorphism of A V with (A V) . The pairing between these two spaces also takes the form of an inner product. On decomposable &-multivectors, (vi A ■■ ■ Av k ,wi A ■ ■■ Aiujt) = det((vi, Wj)), the determinant of the matrix of inner products. In the special case v.-w., the inner product is the square norm of the multi vector, given by the determinant of the Gramian matrix (Dv., v.D). This is then extended bilinearly (or ' J k sesquilinearly in the complex case) to a non-degenerate inner product on A V. If e., i-l,2,...,n, form an orthonormal basis of V, then the vectors of the form e h A---Ae ifc , i a < ■ • ■ < i k , k constitute an orthonormal basis for A (V). Grassmann algebra 168 With respect to the inner product, exterior multiplication and the interior product are mutually adjoint. Specifically, for v € A k ~ l (V), w € A*(V), and x € V, <zAv,w) = (v,vw> b * where x € V is the linear functional defined by x\y) = {x y y) for all y £ V. This property completely characterizes the inner product on the exterior algebra. Functoriality Suppose that V and W are a pair of vector spaces and/ : V — > W is a linear transformation. Then, by the universal construction, there exists a unique homomorphism of graded algebras A(/) : A(V) -> A(W) such that A(/)| AW =f:V = A^V) ^ W = A 1 ^). In particular, A(f) preserves homogeneous degree. The fe-graded components of A(f) are given on decomposable elements by A(/)(x x A . . . A x k ) = /(ari) A ... A f(x k ). Let A k (f)=A(f) AHv) :A k (V)^A k (W). The components of the transformation A(k) relative to a basis of V and W is the matrix of k x fc minors of /. In particular, if V = W and V is of finite dimension n, then A"(/) is a mapping of a one-dimensional vector space A" to itself, and is therefore given by a scalar: the determinant off. Exactness if O^U^V ->W ->0 is a short exact sequence of vector spaces, then -> A^C/) A A(V) -» A(V) -> A(W) -> is an exact sequence of graded vector spaces as is -> A(!7) -> A(V). nil Direct sums In particular, the exterior algebra of a direct sum is isomorphic to the tensor product of the exterior algebras: A(V ®W)= A(V) ® A(W). This is a graded isomorphism; i.e., A k (V © W) = A P (V) ® A*(W). p+q=k Slightly more generally, if O^U^V ->W ->0 is a short exact sequence of vector spaces then A (V) has a filtration = F° C F 1 C ■ ■ ■ C F k C F fe+1 = A*(K) with quotients : _f 1 P+ 1 /_pp = A fc_p (t/) ® A P (H^)- In particular, if 1/ is 1-dimensional then Grassmann algebra 169 -► U ® A* -1 ^) -► A k (V) ->■ A fe (W) -> is exact, and if Wis 1 -dimensional then -> A fe (£f) -> A fe {I/) -> A*" 1 ^) ® W -> .,[12] The alternating tensor algebra ri3i If K is a field of characteristic 0, then the exterior algebra of a vector space V can be canonically identified with the vector subspace of T(V) consisting of antisymmetric tensors. Recall that the exterior algebra is the quotient of T(V) by the ideal / generated by x ® x. Let T (V) be the space of homogeneous tensors of degree r. This is spanned by decomposable tensors Vi ® . . . ® v r , Vi G V. The antisymmetrization (or sometimes the skew-symmetrization) of a decomposable tensor is defined by Alt(ui ®---®v r ) = — 2J sgnfo-)^!) ® ■ ■ ■ ® W CT (, f7e©r where the sum is taken over the symmetric group of permutations on the symbols { l,...,r}. This extends by linearity and homogeneity to an operation, also denoted by Alt, on the full tensor algebra T(V). The image Alt(T(V)) is the alternating tensor algebra, denoted A(V). This is a vector subspace of T(V), and it inherits the structure of a graded vector space from that on T(V). It carries an associative graded product & defined by t®s = Alt(i® s). Although this product differs from the tensor product, the kernel of Alt is precisely the ideal / (again, assuming that K has characteristic 0), and there is a canonical isomorphism A(V) = \(V). Index notation Suppose that V has finite dimension n, and that a basis e , ..., e of Vis given, then any alternating tensor t € A (V) C 7^(V) can be written in index notation as „ H ^ e i2 ® ■ ■ ■ <g> e ir , where f'l " V is completely antisymmetric in its indices. The wedge product of two alternating tensors t and s of ranks r and p is given by itgs = \^ S gn(<r)f ff < 1 >-" iff Ms i ' r < r + 1 >-" i ' r < r +''>ei 1 ® e i2 ® ■ ■ ■ <g> e, The components of this tensor are precisely the skew part of the components of the tensor product s <E> f, denoted by square brackets on the indices: (t®s) il '" i ' r+p = £[*!■■■*'- s 'H-i-v+i>] The interior product may also be described in index notation as follows. Let £ — ^on-v-ibe an antisymmetric tensor of rank r. Then, for a£ V , i t is an alternating tensor of rank r-l, given by a n where n is the dimension of V. ■r+p Grassmann algebra 170 Applications Linear geometry The decomposable fc-vectors have geometric interpretations: the bivector u A V represents the plane spanned by the vectors, "weighted" with a number, given by the area of the oriented parallelogram with sides u and v. Analogously, the 3-vector u A V A ^represents the spanned 3-space weighted by the volume of the oriented parallelepiped with edges m, v, and w. Projective geometry Decomposable fe-vectors in A V correspond to weighted fc-dimensional subspaces of V. In particular, the Grassmannian of ^-dimensional subspaces of V, denoted Gr . (V), can be naturally identified with an algebraic k k subvariety of the projective space P(A V). This is called the Pliicker embedding. Differential geometry The exterior algebra has notable applications in differential geometry, where it is used to define differential forms. A differential form at a point of a differentiable manifold is an alternating multilinear form on the tangent space at the point. Equivalently, a differential form of degree & is a linear functional on the fe-th exterior power of the tangent space. As a consequence, the wedge product of multilinear forms defines a natural wedge product for differential forms. Differential forms play a major role in diverse areas of differential geometry. In particular, the exterior derivative gives the exterior algebra of differential forms on a manifold the structure of a differential algebra. The exterior derivative commutes with pullback along smooth mappings between manifolds, and it is therefore a natural differential operator. The exterior algebra of differential forms, equipped with the exterior derivative, is a differential complex whose cohomology is called the de Rham cohomology of the underlying manifold and plays a vital role in the algebraic topology of differentiable manifolds. Representation theory In representation theory, the exterior algebra is one of the two fundamental Schur functors on the category of vector spaces, the other being the symmetric algebra. Together, these constructions are used to generate the irreducible representations of the general linear group; see fundamental representation. Physics The exterior algebra is an archetypal example of a superalgebra, which plays a fundamental role in physical theories pertaining to fermions and supersymmetry. For a physical discussion, see Grassmann number. For various other applications of related ideas to physics, see superspace and supergroup (physics). Lie algebra homology Let L be a Lie algebra over a field k, then it is possible to define the structure of a chain complex on the exterior algebra of L. This is a fe-linear mapping d : A P+1 L -> A P L defined on decomposable elements by d{xi A- ■ -A:Ep + i) = -^2(—iy +£+1 [xj,Xe]AxiA- - -AotjA- ■ -Ai^A- ■ -Ax p+1 . The Jacobi identity holds if and only if 33 = 0, and so this is a necessary and sufficient condition for an anticommutative nonassociative algebra L to be a Lie algebra. Moreover, in that case AL is a chain complex with boundary operator 3. The homology associated to this complex is the Lie algebra homology. Grassmann algebra 171 Homological algebra The exterior algebra is the main ingredient in the construction of the Koszul complex, a fundamental object in homological algebra. History The exterior algebra was first introduced by Hermann Grassmann in 1844 under the blanket term of ri4i Ausdehnungslehre, or Theory of Extension. This referred more generally to an algebraic (or axiomatic) theory of extended quantities and was one of the early precursors to the modern notion of a vector space. Saint- Venant also published similar ideas of exterior calculus for which he claimed priority over Grassmann. The algebra itself was built from a set of rules, or axioms, capturing the formal aspects of Cayley and Sylvester's theory of multivectors. It was thus a calculus, much like the propositional calculus, except focused exclusively on the task of formal reasoning in geometrical terms. In particular, this new development allowed for an axiomatic characterization of dimension, a property that had previously only been examined from the coordinate point of view. ri7i The import of this new theory of vectors and multivectors was lost to mid 19th century mathematicians, until being thoroughly vetted by Giuseppe Peano in 1888. Peano's work also remained somewhat obscure until the turn of the century, when the subject was unified by members of the French geometry school (notably Henri Poincare, Elie Cartan, and Gaston Darboux) who applied Grassmann's ideas to the calculus of differential forms. A short while later, Alfred North Whitehead, borrowing from the ideas of Peano and Grassmann, introduced his universal algebra. This then paved the way for the 20th century developments of abstract algebra by placing the axiomatic notion of an algebraic system on a firm logical footing. Notes [I] Provided the characteristic is different from 2. [2] Grassmann (1844) introduced these as extended algebras (cf. Clifford 1878). He used the word dufiere (literally translated as outer, or exterior) only to indicate the produkt he defined, which is nowadays conventionally called exterior product, probably to distinguish it from the outer product as defined in modern linear algebra. [3] This axiomatization of areas is due to Leopold Kronecker and Karl Weierstrass; see Bourbaki (1989, Historical Note). For a modern treatment, see MacLane & Birkhoff (1999, Theorem IX. 2. 2). For an elementary treatment, see Strang (1993, Chapter 5). [4] This definition is a standard one. See, for instance, MacLane & Birkhoff (1999). [5] A proof of this can be found in more generality in Bourbaki (1989). [6] See Sternberg (1964, §111.6). [7] See Bourbaki (1989, III. 7.1), and MacLane & Birkhoff (1999, Theorem XVI. 6. 8). More detail on universal properties in general can be found in MacLane & Birkhoff (1999, Chapter VI), and throughout the works of Bourbaki. [8] Some conventions, particularly in physics, define the wedge product as L) AT} — Alt([J ® Tj). This convention is not adopted here, but is discussed in connection with alternating tensors. [9] Indeed, the exterior algebra of V is the enveloping algebra of the abelian Lie superalgebra structure on V. [10] This part of the statement also holds in greater generality if Vand Ware modules over a commutative ring: That A converts epimorphisms to epimorphisms. See Bourbaki (1989, Proposition 3, III. 7. 2). [II] This statement generalizes only to the case where Vand Ware projective modules over a commutative ring. Otherwise, it is generally not the case that A converts monomorphisms to monomorphisms. See Bourbaki (1989, Corollary to Proposition 12, III. 7. 9). [12] Such a filtration also holds for vector bundles, and projective modules over a commutative ring. This is thus more general than the result quoted above for direct sums, since not every short exact sequence splits in other abelian categories. [13] See Bourbaki (1989, III. 7. 5) for generalizations. [14] Kannenberg (2000) published a translation of Grassmann's work in English; he translated Ausdehnungslehre as Extension Theory. [15] J Itard, Biography in Dictionary of Scientific Biography (New York 1970-1990). [16] Authors have in the past referred to this calculus variously as the calculus of extension (Whitehead 1898; Forder 1941), or extensive algebra (Clifford 1878), and recently as extended vector algebra (Browne 2007). [17] Bourbaki 1989, p. 661. Grassmann algebra 172 References Mathematical references • Bishop, R.; Goldberg, S.I. (1980), Tensor analysis on manifolds, Dover, ISBN 0-486-64039-6 Includes a treatment of alternating tensors and alternating forms, as well as a detailed discussion of Hodge duality from the perspective adopted in this article. • Bourbaki, Nicolas (1989), Elements of mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9 This is the main mathematical reference for the article. It introduces the exterior algebra of a module over a commutative ring (although this article specializes primarily to the case when the ring is a field), including a discussion of the universal property, functoriality, duality, and the bialgebra structure. See chapters III. 7 and III. 1 1 . • Bryant, R.L.; Chern, S.S.; Gardner, R.B.; Goldschmidt, H.L.; Griffiths, P.A. (1991), Exterior differential systems, Springer-Verlag This book contains applications of exterior algebras to problems in partial differential equations. Rank and related concepts are developed in the early chapters. • MacLane, S.; Birkhoff, G (1999), Algebra, AMS Chelsea, ISBN 0-8218-1646-2 Chapter XVI sections 6-10 give a more elementary account of the exterior algebra, including duality, determinants and minors, and alternating forms. • Sternberg, Shlomo (1964), Lectures on Differential Geometry, Prentice Hall Contains a classical treatment of the exterior algebra as alternating tensors, and applications to differential geometry. Historical references • Bourbaki, Nicolas (1989), "Historical note on chapters II and III", Elements of mathematics, Algebra I, Springer-Verlag • Clifford, W. (1878), "Applications of Grassmann's Extensive Algebra" (http://jstor.org/stable/2369379), American Journal of Mathematics (The Johns Hopkins University Press) 1 (4): 350—358, doi: 10.2307/2369379 • Forder, H. G (1941), The Calculus of Extension, Cambridge University Press • Grassmann, Hermann (1844), Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik (http://books. google, com/books ?id=bKgAAAAAMAAJ&pg=PAl&dq=Die+Lineale+Ausdehnungslehre+ein+neuer+ Zweig+der+Mathematik) (The Linear Extension Theory - A new Branch of Mathematics) alternative reference (http://resolver.sub.uni-goettingen.de/purl7PPN534901565) • Kannenberg, Llyod (2000), Extension Theory (translation of Grassmann's Ausdehnungslehre), American Mathematical Society, ISBN 0821820311 • Peano, Giuseppe (1888), Calcolo Geometrico secondo I Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni delta Logica Deduttiva; Kannenberg, Lloyd (1999), Geometric calculus: according to the Ausdehnungslehre of H. Grassmann, Birkhauser, ISBN 978-0817641269. • Whitehead, Alfred North (1898), A Treatise on Universal Algebra, with Applications (http://historical. library. Cornell, edu/cgi-bin/cul. math/docviewer?did=0 1 95000 1 & seq=5), Cambridge Grassmann algebra 173 Other references and further reading • Browne, J.M. (2007), Grassmann algebra - Exploring applications of Extended Vector Algebra with Mathematica, Published on line (http://www.grassmannalgebra.info/grassmannalgebra/book/index.htm) An introduction to the exterior algebra, and geometric algebra, with a focus on applications. Also includes a history section and bibliography. • Spivak, Michael (1965), Calculus on manifolds, Addison- Wesley, ISBN 978-0805390216 Includes applications of the exterior algebra to differential forms, specifically focused on integration and Stokes's theorem. The notation A V in this text is used to mean the space of alternating fe-forms on V; i.e., for Spivak A V is what this article would call A V*. Spivak discusses this in Addendum 4. • Strang, G. (1993), Introduction to linear algebra, Wellesley-Cambridge Press, ISBN 978-0961408855 Includes an elementary treatment of the axiomatization of determinants as signed areas, volumes, and higher-dimensional volumes. • Onishchik, A.L. (2001), "Exterior algebra" (http://eom.springer.de/E/e037080.htm), in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104 • Wendell H. Fleming (1965) Functions of Several Variables, Addison-Wesley. Chapter 6: Exterior algebra and differential calculus, pages 205-38. This textbook in multivariate calculus introduces the exterior algebra of differential forms adroitly into the calculus sequence for colleges. • Winitzki, S. (2010), Linear Algebra via Exterior Products, Published on line (http://sites.google.com/site/ winitzki/linalg) An introduction to the coordinate-free approach in basic finite-dimensional linear algebra, using exterior products. Supergroup 174 Supergroup Supergroup or super group may refer to: Supergroup (music), a music group formed by artists who are already notable or respected in their fields Supergroup (physics), a generalization of groups, used in the study of supersymmetry Supergroup (City of Heroes), the term for player guilds in the City of Heroes MMORPG SuperGroup pic, a British company Supergroup (TV series), a VH1 reality show Supergroup, a geological unit Super-group, a team of superheroes who work together Supergroup, a rarely used term in mathematics for the counterpart of a subgroup In L-carrier, a multiplexed group of Channel Groups Superalgebra In mathematics and theoretical physics, a superalgebra is a Z -graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. The prefix super- comes from the theory of supersymmetry in theoretical physics. Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry where they enter into the definitions of graded manifolds, supermanifolds and superschemes. Formal definition Let K be a fixed commutative ring. In most applications, K is a field such as R or C. A superalgebra over K is a ^-module A with a direct sum decomposition A = A e Ai together with a bilinear multiplication AxA^A such that where the subscripts are read modulo 2. A superring, or Z -graded ring, is a superalgebra over the ring of integers Z. The elements of A. are said to be homogeneous. The parity of a homogeneous element x, denoted by Ixl, is or 1 according to whether it is in A or A . Elements of parity are said to be even and those of parity 1 to be odd. If x and v are both homogeneous then so is the product xy and \xy\ = \x\ + \y\ . An associative superalgebra is one whose multiplication is associative and a unital superalgebra is one with a multiplicative identity element. The identity element in a unital superalgebra is necessarily even. Unless otherwise specified, all superalgebras in this article are assumed to be associative and unital. A commutative superalgebra is one which satisfies a graded version of commutativity. Specifically, A is commutative if yx = (-ljHMzy for all homogeneous elements x and y of A. Superalgebra 175 Examples Any algebra over a commutative ring K may be regarded as a purely even superalgebra over K; that is, by taking A to be trivial. Any Z or N-graded algebra may be regarded as superalgebra by reading the grading modulo 2. This includes examples such as tensor algebras and polynomial rings over K. In particular, any exterior algebra over K is a superalgebra. The exterior algebra is the standard example of a supercommutative algebra. The symmetric polynomials and alternating polynomials together form a superalgebra, being the even and odd parts, respectively. Note that this is a different grading from the grading by degree. Clifford algebras are superalgebras. They are generally noncommutative. The set of all endomorphisms (both even and odd) of a super vector space forms a superalgebra under composition. The set of all square supermatrices with entries in K forms a superalgebra denoted by M (K). This algebra may be identified with the algebra of endomorphisms of a free supermodule over K of rank p\q. Lie superalgebras are a graded analog of Lie algebras. Lie superalgebras are nonunital and nonassociative; however, one may construct the analog of a universal enveloping algebra of a Lie superalgebra which is a unital, associative superalgebra. Further definitions and constructions A superalgebra is an algebra with a ^grading ("even" and "odd" elements) such that (i) the bracket of two generators is always antisymmetric except for two odd elements where it is symmetric and (ii) the Jacobi identities are satisfied. [E i3 {0 k , b }} = {[Ei, fe ], O b } + {[Ei, OJ, fc } [0*, {O b , O a }] = [{O k , O k }, O a ] + [{0 fc , 0„}, b ] The first of these three identities says that the form a representation of the ordinary Lie algebra spanned by E (Consider the as vectors on which the E act.) The second is equivalent to the first if the Killing form is nonsingular. The last identity restricts the possible representations of the ordinary Lie algebra. This relation is the reason that not every ordinary Lie algebra can be extended to a superalgebra. Even subalgebra Let A be a superalgebra over a commutative ring K. The submodule A , consisting of all even elements, is closed under multiplication and contains the identity of A and therefore forms a subalgebra of A, naturally called the even subalgebra. It forms an ordinary algebra over K. The set of all odd elements A is an A -bimodule whose scalar multiplication is just multiplication in A. The product in A equips A with a bilinear form H : Aj ® Ao A ± -> A such that fi(x ®y) ■ z — x ■ fi(y ® z) for all x, v, and z in A . This follows from the associativity of the product in A. Superalgebra 176 Grade involution There is a canonical involutive automorphism on any superalgebra called the grade involution. It is given on homogeneous elements by x = (-l)Wx and on arbitrary elements by X = Xq — X\ where x. are the homogeneous parts of x. If A has no 2-torsion (in particular, if 2 is invertible) then the grade involution can be used to distinguish the even and odd parts of A: Ai = {x€A:x = (-l)Vj-. Supercommutativity The supercommutator on A is the binary operator given by [x,y]=xy-{-l)WMyx on homogeneous elements. This can be extended to all of A by linearity. Elements x and y of A are said to supercommute if [x, y] = 0. The supercenter of A is the set of all elements of A which supercommute with all elements of A: Z(A) = {aeA:[fl,i]=0 for all x e A}. The supercenter of A is, in general, different than the center of A as an ungraded algebra. A commutative superalgebra is one whose supercenter is all of A. Super tensor product The graded tensor product of two superalgebras may be regarded as a superalgebra with a multiplication rule determined by: (ai <g> 6i)(o2 <g> 62) = (-l) |bl||a2| (aia 2 ® 6162). Generalizations and categorical definition One can easily generalize the definition of superalgebras to include superalgebras over a commutative superring. The definition given above is then a specialization to the case where the base ring is purely even. Let R be a commutative superring. A superalgebra over R is a 7?-supermodule A with a 7?-bilinear multiplication A x A ^ A that respects the grading. Bilinearity here means that r ■ (xy) = (r ■ x)y = {—V^^x^ ■ y) for all homogeneous elements r € R and x, y G A. Equivalently, one may define a superalgebra over R as a superring A together with an superring homomorphism R — > A whose image lies in the supercenter of A. One may also define superalgebras categorically. The category of all 7?-supermodules forms a monoidal category under the super tensor product with R serving as the unit object. An associative, unital superalgebra over R can then be defined as a monoid in the category of 7?-supermodules. That is, a superalgebra is an 7?-supermodule A with two (even) morphisms fj, : A® A —> A ■q:R-> A for which the usual diagrams commute. Superalgebra 177 Notes [1] Kac, Martinez & Zelmanov (2001), p. 3 (http://books. google. com/books?id=jTCNZz2Tk4cC&pg=PA3&dq="superalgebra"). [2] P. van Nieuwenhuizen, Phys. Rep. 68, 189 (1981) References • Deligne, Pierre; John W. Morgan (1999). "Notes on Supersymmetry (following Joseph Bernstein)". Quantum Fields and Strings: A Course for Mathematicians . 1. American Mathematical Society, pp. 41—97. ISBN 0-8218-2012-5. • Manin, Y. I. (1997). Gauge Field Theory and Complex Geometry ((2nd ed.) ed.). Berlin: Springer. ISBN 3-540-61378-1. • Varadarajan, V. S. (2004). Supersymmetry for Mathematicians: An Introduction. Courant Lecture Notes in Mathematics 11. American Mathematical Society. ISBN 0-8218-3574-2. • Kac, Victor G.; Martinez, Consuelo; Zelmanov, Efim (2001). Graded simple Jordan superalgebras of growth one. Memoirs of the AMS Series. 711. AMS Bookstore. ISBN 9780821826454. Algebroid In mathematics, algebroid may mean • algebroid branch, a formal power series branch of an algebraic curve • algebroid multifunction • Lie algebroid in the theory of Lie groupoids • algebroid cohomology 178 Algebraic Geometry Algebraic geometry Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find some solution; this leads into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique. This Togliatti surface is an algebraic surface of degree five. The fundamental objects of study in algebraic geometry are algebraic varieties, geometric manifestations of solutions of systems of polynomial equations. Plane algebraic curves, which include lines, circles, parabolas, lemniscates, and Cassini ovals, form one of the best studied classes of algebraic varieties. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve relative position of different curves and relations between the curves given by different equations. Descartes's idea of coordinates is central to algebraic geometry, but it has undergone a series of remarkable transformations beginning in the early 19th century. Before then, the coordinates were assumed to be tuples of real numbers, but this changed when first complex numbers, and then elements of an arbitrary field became acceptable. Homogeneous coordinates of projective geometry offered an extension of the notion of coordinate system in a different direction, and enriched the scope of algebraic geometry. Much of the development of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on 'intrinsic' properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology and complex geometry. One key distinction between classical projective geometry of 19th century and modern algebraic geometry, in the form given to it by Grothendieck and Serre, is that the former is concerned with the more geometric notion of a point, while the latter emphasizes the more analytic concepts of a regular function and a regular map and extensively draws on sheaf theory. Another important difference lies in the scope of the subject. Grothendieck' s idea of scheme provides the language and the tools for geometric treatment of arbitrary commutative rings and, in particular, bridges algebraic geometry with algebraic number theory. Andrew Wiles's celebrated proof of Fermat's last theorem is a vivid testament to the power of this approach. Andre Weil, Grothendieck, and Deligne also demonstrated that the fundamental ideas of topology of manifolds have deep analogues in algebraic geometry over finite fields. Algebraic geometry 179 Zeros of simultaneous polynomials In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that simultaneously satisfy one or more polynomial equations. For instance, the two-dimensional sphere in 3 three-dimensional Euclidean space R could be defined as the set of all points (x,y,z) with * 2 + y 2 + z 2 = 1 x+y+z=0 Sphere and slanted circle 2 , 2,2 x +y + z 1 = 0. A "slanted" circle in R can be defined as the set of all points (x,y,z) which satisfy the two polynomial equations x 2 + y 2 + z 2 x + y + z = 0. 1 = 0, Affine varieties First we start with a field k. In classical algebraic geometry, this field was always the complex numbers C, but many of the same results are true if we assume only that k is algebraically closed. We define A (k) (or more simply A", when k is clear from the context), called the affine n-space over k, to be lc . The purpose of this apparently superfluous notation is to emphasize that one 'forgets' the vector space structure that lc carries. Abstractly speaking, A" is, for the moment, just a collection of points. n 1 A function/ : A — > A is said to be regular if it can be written as a polynomial, that is, if there is a polynomial p in k[x ,...,x ] such that/(f ,...,? )=p{t .,...,? ) for every point (t ,...,t ) of A". Regular functions on affine n-space are thus exactly the same as polynomials over k in n variables. We will refer to the set of all regular functions on A" as k[A\. We say that a polynomial vanishes at a point if evaluating it at that point gives zero. Let S be a set of polynomials in k[A ]. The vanishing set of S (or vanishing locus) is the set V(S) of all points in A' where every polynomial in S vanishes. In other words, V{S) = {fa, . . . , t n )\Vp G S,p{t 1: ...,t n ) = 0}. A subset of A ' which is V(S), for some S, is called an algebraic set. The V stands for variety (a specific type of algebraic set to be defined below). Given a subset U of A , can one recover the set of polynomials which generate it? If U is any subset of A , define I(U) to be the set of all polynomials whose vanishing set contains U. The / stands for ideal: if two polynomials /and g both vanish on U, then f+g vanishes on U, and if h is any polynomial, then hf vanishes on U, so I(U) is always an ideal of k[A n ]. Two natural questions to ask are: • Given a subset U of A", when is U= V(I(U))1 Algebraic geometry 180 • Given a set S of polynomials, when is S = I(V(S))1 The answer to the first question is provided by introducing the Zariski topology, a topology on A which directly reflects the algebraic structure of k[A n ]. Then U = V(I(U)) if and only if U is a Zariski-closed set. The answer to the second question is given by Hilbert's Nullstellensatz. In one of its forms, it says that I(V(S)) is the prime radical of the ideal generated by S. In more abstract language, there is a Galois connection, giving rise to two closure operators; they can be identified, and naturally play a basic role in the theory; the example is elaborated at Galois connection. For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set U. Hilbert's basis theorem implies that ideals in k[A n ] are always finitely generated. An algebraic set is called irreducible if it cannot be written as the union of two smaller algebraic sets. An irreducible algebraic set is also called a variety. It turns out that an algebraic set is a variety if and only if the polynomials defining it generate a prime ideal of the polynomial ring. Regular functions Just as continuous functions are the natural maps on topological spaces and smooth functions are the natural maps on differentiable manifolds, there is a natural class of functions on an algebraic set, called regular functions. A regular function on an algebraic set V contained in A n is defined to be the restriction of a regular function on A n , in the sense we defined above. It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a normal topological space, where the Tietze extension theorem guarantees that a continuous function on a closed subset always extends to the ambient topological space. Just as with the regular functions on affine space, the regular functions on V form a ring, which we denote by k[V\. This ring is called the coordinate ring of V. Since regular functions on V come from regular functions on A n , there should be a relationship between their coordinate rings. Specifically, to get a function in k[V\ we took a function in k[A ], and we said that it was the same as another function if they gave the same values when evaluated on V. This is the same as saying that their difference is zero on V. From this we can see that k[V\ is the quotient k[A ]/I(V). The category of affine varieties Using regular functions from an affine variety to A , we can define regular functions from one affine variety to another. First we will define a regular function from a variety into affine space: Let V be a variety contained in A . Choose m regular functions on V, and call them/ , ...,/ . We define a regular function / from V to A by letting f(f,, ..., t ) = (f,, ..., f ). In other words, each f determines one coordinate of the range of f. J 1 n v l J m J i o j If V is a variety contained in A , we say that/is a regular function from V to V if the range of/is contained in V. This makes the collection of all affine varieties into a category, where the objects are affine varieties and the morphisms are regular maps. The following theorem characterizes the category of affine varieties: The category of affine varieties is the opposite category to the category of finitely generated integral fc-algebras and their homomorphisms. Algebraic geometry 181 Projective space 2 Consider the variety V(y - x ). If we draw it, we get a parabola. As x increases, the slope 2 of the line from the origin to the point (x, x ) becomes larger and larger. As x decreases, the slope of the same line becomes smaller and smaller. 3 Compare this to the variety V(y - x ). This is a cubic equation. As x increases, the slope 3 of the line from the origin to the point (x, x ) becomes larger and larger just as before. But unlike before, as x decreases, the slope of the same line again becomes larger and larger. So the behavior "at infinity" of 3 V(y - x') is different from the behavior "at 2 infinity" of V(y - x ). It is, however, difficult to make the concept of "at infinity" meaningful, if we restrict to working in affine space. The remedy to this is to work in projective space. Projective space has properties analogous to those of a compact Hausdorff space. Among other things, it lets us make precise the notion of "at infinity" by including extra points. The 3 behavior of a variety at those extra points then gives us more information about it. As it turns out, V(y - x ) has a 2 singularity at one of those extra points, but V(y - x ) is smooth. While projective geometry was originally established on a synthetic foundation, the use of homogeneous coordinates allowed the introduction of algebraic techniques. Furthermore, the introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For example, Bezout's theorem on the number of intersection points between two varieties can be stated in its sharpest form only in projective space. For this reason, projective space plays a fundamental role in algebraic geometry. 2 3 parabola (y = x , red) and cubic (y = x , blue) in projective space The modern viewpoint The modern approaches to algebraic geometry redefine and effectively extend the range of basic objects in various levels of generality to schemes, formal schemes, ind-schemes, algebraic spaces, algebraic stacks, derived algebraic stacks and so on. The need for this arises already from the useful ideas within theory of varieties, e.g. the formal functions of Zariski can be accommodated by introducing nilpotent elements in structure rings; considering spaces of loops and arcs, constructing quotients by group actions and developing formal grounds for natural intersection theory and deformation theory lead to some of the further extensions. Most remarkably, in late 1950-s, algebraic varieties are subsumed in Alexander Grothendiecks concept of a scheme. Their local objects are affine schemes or prime spectra which are locally ringed spaces which form a category which is antiequivalent to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a field k, and the category of finitely generated reduced Ar-algebras. The gluing is along Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes. The Zariski topology in the set theoretic sense is then replaced by a Zariski topology in the sense of Grothendieck topology. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and more sensitive examples than the crude Zariski topology, namely the etale topology, and the two flat Grothendieck topologies: ffpf and fpqc; nowadays some other examples became prominent including Nisnevich topology. Sheaves can be furthermore Algebraic geometry 182 generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions leading to Artin stacks and, even finer, Deligne-Mumford stacks, both often called algebraic stacks. Sometimes other algebraic sites replace the category of affine schemes. For example, Nikolai Durov has introduced commutative algebraic monads as a generalization of local objects in a generalized algebraic geometry. Versions of a tropical geometry, of an absolute geometry over a field of one element and an algebraic analogue of Arakelov's geometry were realized in this setup. Another formal generalization is possible to Universal algebraic geometry in which every variety of algebra has its own algebraic geometry. The term variety of algebra should not be confused with algebraic variety. The language of schemes, stacks and generalizations has proved to be a valuable way of dealing with geometric concepts and became cornerstones of modern algebraic geometry. Derived algebraic geometry Algebraic stacks can be further generalized and for many practical questions like deformation theory and intersection theory, this is often the most natural approach. One can extend the Grothendieck site of affine schemes to a higher categorical site of derived affine schemes, by replacing the commutative rings with an infinity category of differential graded commutative algebras, or of simplicial commutative rings or a similar category with an appropriate variant of a Grothendieck topology. One can also replace presheaves of sets by presheaves of simplicial sets (or of infinity groupoids). Then, in presence of an appropriate homotopic machinery one can develop a notion of derived stack as such a presheaf on the infinity category of derived affine schemes, which is satifsying certain infinite categorical version of a sheaf axiom (and to be algebraic, inductively a sequence of representability conditions). Quillen model categories, Segal categories and quasicategories are some of the most often used tools to formalize this yielding the derived algebraic geometry, introduced by the school of Carlos Simpson, including Andre Hirschowitz, Bertrand Toen, Gabrielle Vezzosi, Michel Vaquie and others; and recently systematized and applied by Jacob Lurie. Another (noncommutative) version of derived algebraic geometry, using A-infinity categories has been developed from early 1990-s by Maxim Kontsevich and followers. History Prehistory: Before the 19th century Some of the roots of algebraic geometry date back to the work of the Hellenistic Greeks from the 5th century BC. The Delian problem, for instance, was to construct a length x so that the cube of side x contained the same volume as 2 the rectangular box a b for given sides a and b. Menechmus (circa 350 BC) considered the problem geometrically by intersecting the pair of plane conies ay = x and xy = ab. The later work, in the 3rd century BC, of Archimedes and Apollonius studied more systematically problems on conic sections, and also involved the use of coordinates. The Arab mathematicians were able to solve by purely algebraic means certain cubic equations, and then to interpret the results geometrically. This was done, for instance, by Ibn al-Haytham in the 10th century AD. Subsequently, Persian mathematician Omar Khayyam (born 1048 A.D.) discovered the general method of solving cubic equations by intersecting a parabola with a circle. Each of these early developments in algebraic geometry dealt with questions of finding and describing the intersections of algebraic curves. Such techniques of applying geometrical constructions to algebraic problems were also adopted by a number of Renaissance mathematicians such as Gerolamo Cardano and Niccolo Fontana "Tartaglia" on their studies of the cubic equation. The geometrical approach to construction problems, rather than the algebraic one, was favored by most 16th and 17th century mathematicians, notably Blaise Pascal who argued against the use of algebraic and analytical methods in geometry. The French mathematicians Franciscus Vieta and later Rene Descartes and Pierre de Fermat revolutionized the conventional way of thinking about construction problems through the introduction of coordinate geometry. They were interested primarily in the properties of algebraic curves, such as those defined by Algebraic geometry 183 Diophantine equations (in the case of Fermat), and the algebraic reformulation of the classical Greek works on conies and cubics (in the case of Descartes). During the same period, Blaise Pascal and Gerard Desargues approached geometry from a different perspective, developing the synthetic notions of projective geometry. Pascal and Desargues also studied curves, but from the purely geometrical point of view: the analog of the Greek ruler and compass construction. Ultimately, the analytic geometry of Descartes and Fermat won out, for it supplied the 18th century mathematicians with concrete quantitative tools needed to study physical problems using the new calculus of Newton and Leibniz. However, by the end of the 18th century, most of the algebraic character of coordinate geometry was subsumed by the calculus of infinitesimals of Lagrange and Euler. Nineteenth and early 20th century It took the simultaneous 19th century developments of non-Euclidean geometry and Abelian integrals in order to bring the old algebraic ideas back into the geometrical fold. The first of these new developments was seized up by Edmond Laguerre and Arthur Cayley, who attempted to ascertain the generalized metric properties of projective space. Cayley introduced the idea of homogeneous polynomial forms, and more specifically quadratic forms, on projective space. Subsequently, Felix Klein studied projective geometry (along with other sorts of geometry) from the viewpoint that the geometry on a space is encoded in a certain class of transformations on the space. By the end of the 19th century, projective geometers were studying more general kinds of transformations on figures in projective space. Rather than the projective linear transformations which were normally regarded as giving the fundamental Kleinian geometry on projective space, they concerned themselves also with the higher degree birational transformations. This weaker notion of congruence would later lead members of the 20th century Italian school of algebraic geometry to classify algebraic surfaces up to birational isomorphism. The second early 19th century development, that of Abelian integrals, would lead Bernhard Riemann to the development of Riemann surfaces. Twentieth century B. L. van der Waerden, Oscar Zariski, Andre Weil and others attempted to develop a rigorous foundation for algebraic geometry based on contemporary commutative algebra, including valuation theory and the theory of ideals. In the 1950s and 1960s Jean-Pierre Serre and Alexander Grothendieck recast the foundations making use of sheaf theory. Later, from about 1960, and largely spearheaded by Grothendieck, the idea of schemes was worked out, in conjunction with a very refined apparatus of homological techniques. After a decade of rapid development the field stabilized in the 1970s, and new applications were made, both to number theory and to more classical geometric questions on algebraic varieties, singularities and moduli. An important class of varieties, not easily understood directly from their defining equations, are the abelian varieties, which are the projective varieties whose points form an abelian group. The prototypical examples are the elliptic curves, which have a rich theory. They were instrumental in the proof of Fermat's last theorem and are also used in elliptic curve cryptography. While much of algebraic geometry is concerned with abstract and general statements about varieties, methods for effective computation with concretely-given polynomials have also been developed. The most important is the technique of Grobner bases which is employed in all computer algebra systems. Based on these methods, several solvers may compute all the solutions of a system of polynomial equations whose associated variety has dimension zero and thus consists in a finite number of points. Algebraic geometry 1 84 Applications Algebraic geometry now finds application in statistics, control theory, robotics, error-correcting codes, phylogenetics and geometric modelling. There are also connections to string theory, game theory, graph matchings, solitons and integer programming. Google scholar lists hundreds of more studies on algebraic geometry in biology , chemistry , economics , physics and of course other areas of mathematics . Notes [I] Dieudonne, Jean (1972). "The historical development of algebraic geometry" (http://jstor.org/stable/2317664). The American Mathematical Monthly (The American Mathematical Monthly, Vol. 79, No. 8) 79 (8): 827-866. doi: 10.2307/23 17664. [2] Kline, M. (1972) Mathematical Thought from Ancient to Modern Times (Volume 1). Oxford University Press, pp. 108, 90. [3] Kline, M. (1972) Mathematical Thought from Ancient to Modern Times (Volume 1). Oxford University Press, p. 193. [4] Kline, M. (1972) Mathematical Thought from Ancient to Modern Times (Volume 1). Oxford University Press, pp. 193—195. [5] Kline, M. (1972) Mathematical Thought from Ancient to Modern Times (Volume 1). Oxford University Press, p. 279. [6] Mathias Drton, Bernd Sturmfels, Seth Sullivant (2009), Lectures on Algebraic Statistics (http://books.google.co.uk/ books?id=TytYUTy5V_IC) Springer, ISBN 9783764389048 [7] Peter L. Falb (1990), Methods of algebraic geometry in control theory (http://books. google. co.uk/books?id=oJGYPwAACAAJ&), Birkhauser, ISBN 9783764334543 [8] J. M. Selig (205), Geometric fundamentals of robotics (http://books. google. co.uk/books ?id=GUuzEOWilOQC), Springer, ISBN 9780387208749 [9] Michael A. Tsfasman, Serge G. VladuJ, Dmitry Nogin (2007), Algebraic geometric codes: basic notions (http://books.google.co.uk/ books?id=o2sA-wzDBLkC), AMS Bookstore, ISBN 9780821843062 [10] Barry A. Cipra (2007), Algebraic Geometers See Ideal Approach to Biology (http://siam.org/pdf/news/1146.pdf), SIAM News, Volume 40, Number 6 [II] Bert Juttler, Ragni Piene (2007) Geometric modeling and algebraic geometry (http://books. google. co.uk/books 7id=lwNGq87gWykC), Springer, ISBN 9783540721840 [12] David A. Cox, Sheldon Katz (1999) Mirror symmetry and algebraic geometry (http://books. google. co.uk/books?id=vwL4ZewC81MC), AMS Bookstore, ISBN 9780821821275 [13] The algebraic geometry of perfect and sequential equilibrium (http://econwpa.wustl.edu/econ-wp/game/papers/9309/9309001.pdf), LE Blume, WRZame - Econometrica: Journal of the Econometric Society, 1994 -jstor.org [14] Richard Kenyon, Andrei Okounkov, Scott Sheffield (2003) Dimers and Amoebae (http://arxiv.org/abs/math-ph/031 1005vl) [15] IM Krichever and PG Grinevich, Algebraic geometry methods in soliton theory, Chapter 14 of Soliton theory (http://books.google.co.uk/ books?id=eO_PAAAAIAAJ), Allan P. Fordy, Manchester University Press ND, 1990, ISBN 9780719014918 [16] David A. Cox, Bernd Sturmfels, Dinesh N. Manocha (1997 Applications of computational algebraic geometry (http://books.google.co.uk/ books?id=feOMJEPDwzAC), AMS Bookstore, ISBN 9780821807507 [17] http://scholar.google.co. uk/scholar?q=%22Algebraic+Geometry%22&hl=en&num=100&as_subj=bio [18] http://scholar.google.co. uk/scholar?q=%22Algebraic+Geometry%22&hl=en&num=100&as_subj=chm [19] http://scholar.google.co. uk/scholar?q=%22Algebraic+Geometry%22&hl=en&num=100&as_subj=bus [20] http://scholar.google.co. uk/scholar?q=%22Algebraic+Geometry%22&hl=en&num=100&as_subj=phy [21] http://scholar.google.co. uk/scholar?q=%22Algebraic+Geometry%22&hl=en&num=100&as_subj=eng References A classical textbook, predating schemes: • W. V. D. Hodge; Daniel Pedoe (1994). Methods of Algebraic Geometry: Volume 1. Cambridge University Press. ISBN 0-521-46900-7. • W. V. D. Hodge; Daniel Pedoe (1994). Methods of Algebraic Geometry: Volume 2. Cambridge University Press. ISBN 0-521-46901-5. • W. V. D. Hodge; Daniel Pedoe (1994). Methods of Algebraic Geometry: Volume 3. Cambridge University Press. ISBN 0-521-46775-6. Modern textbooks that do not use the language of schemes: • David A. Cox; John Little, Donal O'Shea (1997). Ideals, Varieties, and Algorithms (second ed.). Springer- Verlag. ISBN 0-387-94680-2. • Phillip Griffiths; Joe Harris (1994). Principles of Algebraic Geometry. Wiley-Interscience. ISBN 0-471-05059-8. Algebraic geometry 185 • Joe Harris (1995). Algebraic Geometry: A First Course. Springer- Verlag. ISBN 0-387-97716-3. • David Mumford (1995). Algebraic Geometry I: Complex Projective Varieties (2nd ed.). Springer- Verlag. ISBN 3-540-58657-1. • Miles Reid (1988). Undergraduate Algebraic Geometry. Cambridge University Press. ISBN 0-521-35662-8. • Igor Shafarevich (1995). Basic Algebraic Geometry I: Varieties in Projective Space (2nd ed.). Springer- Verlag. ISBN 0-387-54812-2. Textbooks and references for schemes: • David Eisenbud; Joe Harris (1998). The Geometry of Schemes. Springer- Verlag. ISBN 0-387-98637-5. • Alexander Grothendieck (1960). Elements de geometrie algebrique. Publications mathematiques de 1'IHES. • Alexander Grothendieck (1971). Elements de geometrie algebrique. 1 (2nd ed.). Springer- Verlag. ISBN 3-540-05113-9. • Robin Hartshorne (1997). Algebraic Geometry. Springer- Verlag. ISBN 0-387-90244-9. • David Mumford (1999). The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians (2nd ed.). Springer- Verlag. ISBN 3-540-63293-X. • Igor Shafarevich (1995). Basic Algebraic Geometry II: Schemes and Complex Manifolds. Springer- Verlag. ISBN 0-387-54812-2. On the Internet: • Kevin R. Coombes: Algebraic Geometry: A Total Hypertext Online System (http://odin.mdacc.tmc.edu/~krc/ agathos/). In construction; currently of very limited use for self study. • Algebraic geometry (http://planetmath.org/encyclopedia/AlgebraicGeometry.html) entry on PlanetMath (http:/ /planetmath.org/) • Algebraic Equations and Systems of Algebraic Equations (http://eqworld.ipmnet.ru/en/solutions/ae.htm) at Eq World: The World of Mathematical Equations List of algebraic geometry topics 186 List of algebraic geometry topics This is a list of algebraic geometry topics, by Wikipedia page. Classical topics in projective geometry • Affine space • Projective space • Projective line, cross-ratio • Projective plane • Line at infinity • Complex projective plane Complex projective space Plane at infinity, hyperplane at infinity Projective frame Projective transformation Fundamental theorem of projective geometry Duality (projective geometry) Real projective plane Real projective space Segre embedding, multi-way projective space Rational normal curve Algebraic curves • Conies, Pascal's theorem, Brianchon's theorem • Twisted cubic • Elliptic curve, cubic curve • Elliptic function, Jacobi's elliptic functions, Weierstrass's elliptic functions • Elliptic integral • Complex multiplication • Weil pairing • Hyperelliptic curve • Klein quartic • modular curve • modular equation • modular function • modular group • Supersingular primes Fermat curve Bezout's theorem Brill— Noether theory Edwards curve Genus (mathematics) Riemann surface Riemann— Hurwitz formula Riemann— Roch theorem List of algebraic geometry topics 187 • Abelian integral • Differential of the first kind • Jacobian variety • Generalized Jacobian • Hurwitz's automorphisms theorem • Clifford's theorem • Gonality of an algebraic curve • Weil's reciprocity law • Goppa code Algebraic surfaces Enriques-Kodaira classification List of algebraic surfaces Ruled surface Cubic surface Veronese surface Del Pezzo surface Rational surface Enriques surface K3 surface Hodge index theorem Elliptic surface Surface of general type Zariski surface Algebraic geometry: classical approach • Algebraic variety Hypersurface Quadric Dimension of an algebraic variety Hilbert's Nullstellensatz Complete variety Elimination theory Quasiprojective variety • Grobner basis Canonical bundle Complete intersection Serre duality Arithmetic genus, geometric genus, irregularity Tangent space, Zariski tangent space Function field Ample vector bundle Linear system of divisors Birational geometry • Blowing up List of algebraic geometry topics 188 • Rational variety • Unirational variety • Intersection number • Intersection theory • Serre's multiplicity conjectures Albanese variety Picard group Pluricanonical ring Modular form Moduli space Modular equation • J-invariant Algebraic function Algebraic form Addition theorem Invariant theory • Symbolic method of invariant theory Geometric invariant theory Toric geometry Deformation theory Singular point, non-singular Singularity theory • Newton polygon • Weil conjectures Complex manifolds Kahler manifold Calabi— Yau manifold Stein manifold Hodge theory Hodge cycle Hodge conjecture Algebraic geometry and analytic geometry Mirror symmetry Algebraic groups • Identity component • Linear algebraic group Additive group Multiplicative group Borel subgroup Parabolic subgroup Radical of an algebraic group Unipotent radical Lie-Kolchin theorem List of algebraic geometry topics 189 • Mumford conjecture • Abelian variety • Theta function • Grassmannian • Flag manifold • Algebraic torus • Weil restriction • Differential Galois theory Contemporary foundations Main article glossary of scheme theory Commutative algebra Prime ideal Valuation (mathematics) Regular local ring Regular sequence (algebra) Cohen— Macaulay ring Gorenstein ring Koszul complex Spectrum of a ring Zariski topology Kahler differential Generic flatness Irrelevant ideal Sheaf theory Locally ringed space Coherent sheaf Invertible sheaf Sheaf cohomology Hirzebruch— Riemann— Roch theorem Grothendieck— Riemann— Roch theorem Coherent duality Devissage Schemes Affine scheme Scheme Glossary of scheme theory Elements de geometrie algebrique Grothendieck's Seminaire de geometrie algebrique Flat morphism Finite morphism Quasi-finite morphism Group scheme List of algebraic geometry topics 190 • Semistable elliptic curve • Grothendieck's relative point of view Category theory Grothendieck topology Topos Descent (category theory) • Grothendieck's Galois theory Algebraic stack Gerbe Etale cohomology Motive (mathematics) Motivic cohomology Homotopical algebra Algebraic geometers Niels Henrik Abel Carl Gustav Jakob Jacobi Jakob Steiner Julius Pliicker Bernhard Riemann William Kingdon Clifford Italian school of algebraic geometry • Guido Castelnuovo • Francesco Severi Solomon Lefschetz Oscar Zariski Erich Kahler W. V. D. Hodge Kunihiko Kodaira Andre Weil Jean-Pierre Serre Alexander Grothendieck David Mumford Igor Shafarevich Heisuke Hironaka Shigefumi Mori Vladimir Voevodsky Duality (projective geometry) 191 Duality (projective geometry) In the geometry of the projective plane, duality refers to geometric transformations that replace points by lines and lines by points while preserving incidence properties among the transformed objects. The existence of such transformations leads to a general principle, that any theorem about incidences between points and lines in the projective plane may be transformed into another theorem about lines and points, by a substitution of the appropriate words. Duality in the projective plane is a special case of duality for projective spaces, transformations that interchange dimension + codimension. That is, in a projective space of dimension n, the points (dimension 0) are made to correspond with hyperplanes (codimension 1), the lines joining two points (dimension 1) are made to correspond with the intersection of two hyperplanes (codimension 2), and so on. Duality in terms of vector space The points of n-dimensional projective space over a field F, called CF , can be taken to be the nonzero vectors in the (n + l)-dimensional vector space over F, where we identify two vectors which differ by a scalar factor. Another way to put it is that the points of n-dimensional projective space are the lines through the origin in F" . Also the n-dimensional subspaces of F" represent the (n - l)-dimensional hyperplanes of projective n-space over F. A nonzero vector u in F also determines an n-dimensional subspace, by means of the equation UqXq -\ h u n x n = where u. is the ith coordinate of u, starting from zero, all w.'s are from field F. In terms of the dot product we can write this as u . x = 0> where u is the vector representing a hyperplane, and x the vector representing a point. The dot product is symmetrical, and the same vector u represents both a point and a hyperplane. Hence we have a duality between points and hyperplanes, which extends to a duality between the line generated by two points and the intersection of two hyperplanes, and so forth. Points and lines in the plane Notice that both points and lines can be represented (on a plane) by means of ordered pairs. A point is represented by the ordered pair (x,y), where x is the abscissa and y is the ordinate, which together are coordinates of the point. A line can likewise be represented by an ordered pair (m,b) where m is the slope and b is the y-intercept. Given three points Pi ■ ( x i,yi), P-i ■ (22,2/2), Ps ■ (23,1/3); these three points are collinear (i.e., they lie on the same line) if and only if their coordinates satisfy the equation 2/2 - 2/i 2/3 - 2/2 2/3 - 2/1 x 2 — x 1 x 3 — x 2 x 3 - x 1 Likewise, given three lines (!)• £1 (mij&i), (m 2 ,6 2 ), (™3,&3); Duality (projective geometry) 192 one can verify that these three lines are concurrent (i.e., all share an intersection) if and only if their parameters satisfy the equation h ~bj _ b 3 - b 2 _ b 3 - b x Equations (1) and (2) are equivalent to each other up to an exchange of x with m and y with b. Therefore there exists a way to exchange lines with points in such a way that concurrency is exchanged with collinearity. It is possible to distinguish lines from points by conjugating ordered pairs. That is, let line (m,b) be represented instead by its conjugate (m, —b)*. Then it can be verified that the intersection L .L of a pair of lines L and L is ( u\* ( u \* f b 2~h mi& 2 - m 2 b 1 \ (mi, 6i)*.(m 2 , b 2 f = , (3) \rri2 — mi m 2 — m\ J where b and b are negative y-intercepts. Also, the common line P .P passing through a pair of points P and P is , w x ( V2-yi x 1 y 2 -x 2 y 1 \ k {x 1 ,y 1 ).{x 2 ,y 2 ) = , . (4) \x 2 — xi x 2 — x 1 J Equation (4) can be seen to be the same as equation (3), after exchanging m with x and b with y, and applying the following rules of conjugation: A** = A, (5) A = B -> A* = B*, (6) (A.B)* = A*.B* = B*.A*. (7) Indeed, if equation (3) is represented as A*.B* = C then applying rule (6) yields (A*.£*)* = C*. Applying rule (7) then yields A**.B** = C* and applying rule (5) finally yields A.B = C*, which is equation (4). Thus it is possible to imagine a pair of planes S and S , and a bijective relation between loci of points in the two planes, such that points in S correspond to lines in S , and points in S correspond to lines in S . Great circles One way to establish such bijection is to model the real projective plane, not as an extended affine plane, but as a "unit sphere modulo antipodes", i.e. a unit sphere in which antipodal points are equivalent. Then through points P and P in S passes a geodesic line L which is actually a great circle. But to these two original points correspond a pair of great circles L and L in S , such that if S and S are superposed, then L is the unique great circle perpendicular to the line through the pair of points P.* 1 , and L is the unique great circle perpendicular to the line through the pair of points P . These great circles L and L intersect at a pair of points P in S . The vector through P is the cross product of the vectors through P and P . Then the unique great circle perpendicular to the line passing through the pair of points P is geodesic line L in S . Therefore to every great circle in S corresponds a unique pair of points (which are actually the same point) in S , such that if S and S are superposed, then the (3-D) line passing through the pair of points is perpendicular to (the plane in 3-D of) the great circle. The above sentence remains true if S and S are exchanged. This establishes the bijective nature of the duality in the projective plane. Duality (projective geometry) 193 It must be noted that in the "unit sphere modulo antipodes", one "geodesic line", i.e. great circle, must be chosen to be the line at infinity if the surface is to be mapped to an extended affine plane. This line may be chosen to be the equator by convention. • *' When we say a line through pair of points P, or simply a line through P, we refer to the three dimensional euclidian line that passes through the antipodal points represented by P. When we say that a geodesic line, or a great circle, L is perpendicular to the line passing through the pair of points P, we mean that L lies on the plane that is perpendicular to, and intersects at the midpoint of, the straight line segment in euclidian space that connects the antipodal points that is represented by P. In other words, L is the set of points equidistant in euclidian space to the antipodal points represented by P. L is unique for P. Three dimensions There is also a duality in projective 3-space, in which points correspond to planes, and lines correspond to lines. This is analogous to duality of polyhedra in solid geometry, where points are dual to faces, and sides are dual to sides, so that the icosahedron is dual to the dodecahedron, and the cube is dual to the octahedron. Mapping the sphere onto the plane The unit sphere modulo -1 model of the projective plane is isomorphic (w.r.t. incidence properties) to the planar model: the affine plane extended with a projective line at infinity. To map a point on the sphere to a point on the plane, let the plane be tangent to the sphere at some point which shall be the origin of the plane's coordinate system (2-D origin). Then construct a line passing through the center of the sphere (3-D origin) and the point on the sphere. This line intersects the plane at a point which is the projection of the point on the sphere onto the plane (or vice versa). This projection can be used to define a one-to-one onto mapping /: [0,tt/2] x [0,2tt]^RP 2 . If points in ]g^p 2 are expressed in homogeneous coordinates, then / : (9, (/>) i— > [cos (j) : sin (j) : cot 0], / : [x : y : z] \— > [ arctan \ I — ) + I — J , arctari2(y, x) Also, lines in the planar model are projections of great circles of the sphere. This is so because through any line in the plane pass an infinitude of different planes: one of these planes passes through the 3-D origin, but a plane passing through the 3-D origin intersects the sphere along a great circle. As we have seen, any great circle in the unit sphere has a projective point perpendicular to it, which can be defined as its dual. But this point is a pair of antipodal points on the unit sphere, through both of which passes a unique 3-D line, and this line extended past the unit sphere intersects the tangent plane at a point, which means that there is a geometric way to associate a unique point on the plane to every line on the plane, such that the point is the dual of the line. Duality (projective geometry) 194 Duality mapping defined Given a line L in the projective plane, what is its dual point? Draw a line L' passing through the 2-D origin and perpendicular to line L. Then pick a point P on line L' on the other side of the origin from line L, such that the distance of point P to the origin is the reciprocal of the distance of line L to the origin. Figure 1. Three pairs of dual points and lines: one red pair, one yellow pair, and one blue pair. The duality is an isomorphism of incidence, so that, e.g., the line passing through the red and yellow points is dual to the intersection of the red and yellow lines. Expressed algebraically, let g be a one-to-one mapping from the projective plane onto itself: q : MP 2 -> MP 2 such that g:[m:b:l] L and m : ■1:6] g:[x:y:l]\-*[x:l: -y] L where the L subscript is used to semantically distinguish line coordinates from point coordinates. In words, affine line (m, b) with slope m and y-intercept b is the dual of point (m/b, -lib). If b=0 then the line passes through the 2-D origin and its dual is the ideal point [m : -1 : 0]. The affine point with Cartesian coordinates (x,y) has as its dual the line whose slope is -xly and whose y-intercept is -1/y. If the point is the 2-D origin [0:0:1], then its dual is [0:1:0] which is the line at infinity. If the point is [x:0:l], on the x-axis, then its dual is line [x: 1:0] which shall be interpreted as a line whose slope is vertical and whose x-intercept is -1/x. If a point or a line's homogeneous coordinates are represented as a vector in 3x1 matrix form, then the duality mapping g can be represented as a trilinear transformation, a 3x3 matrix Duality (projective geometry) 195 1 0-1 1 whose inverse is G~ l = Matrix G has one real eigenvalue: one, whose eigenvector is [1:0:0]. The line [1:0:0] is the y-axis, whose dual is the ideal point [1:0:0] which is the intersection of the ideal line with the x-axis. Notice that [1:0:0] is the y-axis, [0:1:0] is the line at infinity, and [0:0:1] is the x-axis. In 3-space, matrix G is a 90° rotation about the x-axis which turns the y-axis into the z-axis. In projective 2-space, matrix G is a projective transformation which maps points to points, lines to lines, conic sections to conic sections: it exchanges the line at infinity with the x-axis and maps the y-axis onto itself through a Mobius transformation. As a duality, matrix G pairs up each projective line with its dual projective point. 1 0" 1 -1 Preservation of incidence The duality mapping g is an isomorphism with respect to the incidence properties (such as collinearity and concurrency). The mapping g has this property: given a pair of lines L and L which intersect at a point P, then their -1 12 dual points gL and gL define the unique line g P: g~ 1 (L 1 C\L 2 ) = gL 1 .gL 2 - Given points P and P through which passes line L, P ' ,P ' = L, then what is the intersection of lines g~ P and g~ l P 2 l If g~ 1 P ] n g~ 1 P 2 = P then g-'P = g-'ig^P, H g~ 1 P 2 ) = g^ 1 P,) .g^ 1 P 2 ) = Pi-Pi = L so that gig-'P) = gL P = gL ■•• g(P 1 .P 2 )=g- 1 P 1 r\g- 1 P 2 Given a pair of affine points in homogeneous coordinates, the line passing through them is [xi : T/i : l].[x 2 : y 2 : 1] = 5 _1 ([xi : y x : 1] X [x 2 : y 2 : 1]) where the cross product is computed just as it would for an ordinary pair vectors in 3-space. From this last equation can be derived the intersection of lines, by using the mapping g to "plug in" the lines into the slots for points: g[mi : b ± : l] L -g[m 2 : b 2 : 1] L = g~ 1 (g[m 1 : &i : 1] L X g[m 2 : b 2 : 1] L ) g(g[mi : b x : l] L -g[m 2 : b 2 : 1] L ) = g[mi : h : 1] L X g[m 2 : b 2 : 1]^ [mi : &i : 1]l n [m 2 : b 2 : 1] L = g([m 1 : bj : 1] L X [m 2 : b 2 : 1]^) where mapping g is seen to distribute with respect to the cross product: i.e. g is an isomorphism of cross product. Theorem. The duality mapping g is an isomorphism of cross product. I.e. g is distributive w.r.t. cross product. Proof. Given points A={a:b:c) and B=(d:e:f), their cross product is (a : b : c) X (d : e : f) — (bf - ce : cd - af : ae - bd) but g(a : & : c) = (a : — c : 6), Duality (projective geometry) 196 g{d:e:f) = {d:-f: e), (a : — c : b) X (d : — / : e) = (— ce + bf : bd — ae : —af + cd) = g{bf — ce : cd — af : ae — bd). Therefore g(A xB)=gAxgB. Q.E.D. Combinatorial duality If one defines a projective plane axiomatically as an incidence structure, in terms of a set P of points, a set L of lines, and an incidence relation / that determines which points lie on which lines, then one may define duality abstractly thus. If we interchange the role of "points" and "lines" in C = (P,L,I), the dual structure is obtained, where /* is the inverse relation of /. References Weisstein, Eric W., "Duality Principle [1] " from MathWorld. References [1] http://mathworld.wolfram.com/DualityPrinciple.html Universal algebraic geometry 197 Universal algebraic geometry In universal algebraic geometry, algebraic geometry is generalized from the geometry of rings to geometry of arbitrary varieties of algebras, so that every variety of algebras has its own algebraic geometry. Note that the two terms algebraic variety and variety of algebras should not be confused. References • Seven Lectures on the Universal Algebraic Geometry References [1] http://arxiv.org/abs/math/0204245 Motive (algebraic geometry) In algebraic geometry, a motive (or sometimes motif, following French usage) denotes 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples (X, p, m), where X is a smooth projective variety, p : X \- X is an idempotent correspondence, and m an integer. A morphism from (X, p, m) to ( Y, q, n) is given by a correspondence of degree n - m. As far as mixed motives, following Alexander Grothendieck, mathematicians are working to find a suitable definition which will then provide a "universal" cohomology theory. In terms of category theory, it was intended to have a definition via splitting idempotents in a category of algebraic correspondences. The way ahead for that definition has been blocked for some decades, by the failure to prove the standard conjectures on algebraic cycles. This prevents the category from having 'enough' morphisms, as can currently be shown. While the category of motives was supposed to be the universal Weil cohomology much discussed in the years 1960-1970, that hope for it remains unfulfilled. On the other hand, by a quite different route, motivic cohomology now has a technically-adequate definition. Introduction The theory of motives was originally conjectured as an attempt to unify a rapidly multiplying array of cohomology theories, including Betti cohomology, de Rham cohomology, /-adic cohomology, and crystalline cohomology. The general hope is that equations like • [point] • [projective line] = [line] + [point] • [projective plane] = [plane] + [line] + [point] can be put on increasingly solid mathematical footing with a deep meaning. Of course, the above equations are already known to be true in many senses, such as in the sense of CW-complex where "+" corresponds to attaching cells, and in the sense of various cohomology theories, where "+" corresponds to the direct sum. From another viewpoint, motives continue the sequence of generalizations from rational functions on varieties to divisors on varieties to Chow groups of varieties. The generalization happens in more than one direction, since motives can be considered with respect to more types of equivalence than rational equivalence. The admissiable equivalences are given by the definition of an adequate equivalence relation. Motive (algebraic geometry) 198 Definition of pure motives The category of pure motives often proceeds in three steps. Below we describe the case of Chow motives Chow(k), where lc is any field. First step: category of (degree 0) correspondences, Corr(k) The objects of Corr(k) are simply smooth projective varieties over k. The morphisms are correspondences. They generalize morphisms of varieties X — > Y, which can be associated with their graphs in X x Y, to fixed dimensional Chow cycles on X x Y. It will be useful to describe correspondences of arbitrary degree, although morphisms in Corr(k) are correspondences of degree 0. In detail, let X and Y be smooth projective varieties, let ^=]X x < ^ e me decomposition of X into connected components, and let d. := dim X.. If r € Z, then the correspondences of degree r from X to Y are Corr r (k)(X,Y) := 04*^(1; x F). i Correspondences are often denoted using the "l-"-notation, e.g., a : X \- Y. For any a D Cor/(X, Y) and /3 D Corr (Y, Z), their composition is defined by cl o 13 := 7r XZx (7T* XY (a) ■ n YZ {j3)) € Corr T+s (X, Z), where the dot denotes the product in the Chow ring (i.e., intersection). Returning to constructing the category Corr(k), notice that the composition of degree correspondences is degree 0. Hence we define morphisms of Corr(k) to be degree correspondences. The association, SmProj(k) — ► Corr{k) F: X i — > X / >-> r, where r D X x Y is the graph off : X —> Y, is a functor. Just like SmProj(k), the category Corr(k) has direct sums ( x®Y:=x\^Y) and tensor products (X <E> Y := X x Y). It is a preadditive category (see the convention for preadditive vs. additive in the preadditive category article.) The sum of morphisms is defined by a + P:=(a,p) G A*(X x X) © A*(Y x Y) ^ A*((X JJy) x (A"]jY)). Second step: category of pure effective Chow motives, Chow e ^(k) The transition to motives is made by taking the pseudo-abelian envelope of Corr(k): Chmv eff (k) := Split(Corr(k)). In other words, effective Chow motives are pairs of smooth projective varieties X and idempotent correspondences a : X \- X, and morphisms are of a certain type of correspondence: Ob(Chow eff (k)) :={(X,a)\(a:X\-X)e Corr(k) such that a o a = a} ■ Mor({X, a ),{Y,/3)):={f:X\-Y\foa = f = l3of}. Composition is the above defined composition of correspondences, and the identity morphism of (X, a) is defined to be a : X \- X. The association, SmProj(k) — *• Corr(k) h: X ^ [X]:=(X,A) X , f .— ► [f]:=T f CXxY Motive (algebraic geometry) 199 where A := [id ] denotes the diagonal of X x X, is a functor. The motive [X] is often called the motive associated to the variety X. As intended, Chow (k) is a pseudo-abelian category. The direct sum of effective motives is given by {[X\, a )®{\Y] i p):=([X]m<* + P)' The tensor product of effective motives is defined by ([X],a)®(\Y\,p) := (XxY,Tv* x a-7r^f3), tt x : (XxY)x(XxY) -^XxX, and tt y : (XxY)x(Xx The tensor product of morphisms may also be defined. Let/ : (X , a ) — > (7 , /? ) and/ : (X , a ) — » (7 , /3 ) be morphisms of motives. Then let y G A (X x 7 ) and y G A (X x 7 ) be representatives of/ and/ . Then h ® h : {X 1: a,) ® (X 2 , a 2 ) h (y 1; ft) ® (y 2) ft), A ® / 2 := 7r* 7l ■ tt 2 * 72 , where i.:l,xl x 7, x 7^ — > X. x 7. are the projections. 112 12 ii r J Third step: category of pure Chow motives, Chow(k) To proceed to motives, we adjoin to Chow (k) a formal inverse (with respect to the tensor product) of a motive called the Lefschetz motive. The effect is that motives become triples instead of pairs. The Lefschetz motive L is L:=(P\A), A:=ptxP a G ^(P 1 xP 1 ). If we define the motive 1, called the trivial Tate motive, by 1 := h(Spec(fc)), then the pleasant equation [P 1 ] = 1 © L holds, since 1 = (P , P x pt). The tensor inverse of the Lefschetz motive is known as the Tate motive, T =17 . Then we define the category of pure Chow motives by Chow(k) := Chow eff (k)[T]. A motive is then a triple (X € SmProj(k), p : X \- X, n G Z) such that p " p = p. Morphisms are given by correspondences / : (X,p,m) -> (Y,q,n), f G Corr"{X,Y) such that fop = f = q of, and the composition of morphisms comes from composition of correspondences. As intended, Chow(k) is a rigid pseudo-abelian category. Other types of motives In order to define an intersection product, cycles must be "movable" so we can intersect them in general position. Choosing an suitable equivalence relation on cycles will guarantee that every pair of cycles has an equivalent pair in general position that we can intersect. The Chow groups are defined using rational equivalence, but other equivalences are possible, and each defines a different sort of motive. Examples of equivalences, from strongest to weakest, are • Rational equivalence • Algebraic equivalence • Smash-nilpotence equivalence (sometimes called Voevodsky equivalence) • Homological equivalence (in the sense of Weil cohomology) • Numerical equivalence The literature occasionally calls every type of pure motive a Chow motive, in which case a motive with respect to algebraic equivalence would be called a Chow motive modulo algebraic equivalence. Motive (algebraic geometry) 200 Mixed motives For a fixed base field k, the category of mixed motives is a conjectural abelian tensor category MM{k), together with a contravariant functor Var(k) -> MM(X) taking values on all varieties (not just smooth projective ones as it was the case with pure motives). This should be such that motivic cohomology defined by &f* u (l, ?) MM coincides with the one predicted by algebraic K-theory, and contains the category of Chow motives in a suitable sense (and other properties). The existence of such a category was conjectured by Beilinson. This category is yet to be constructed. Instead of constructing such a category, it was proposed by Deligne to first construct a category DM having the properties one expects for the derived category D b (MM(k)). Getting MM back from DM would then be accomplished by a (conjectural) motivic t-structure. The current state of the theory is that we do have a suitable category DM. Already this category is useful in applications. Voevodsky's Fields medal-winning proof of the Milnor conjecture uses these motives as a key ingredient. There are different definitions due to Hanamura, Levine and Voevodsky. They are known to be equivalent in most cases and we will give Voevodsky's definition below. The category contains Chow motives as a full subcategory and gives the "right" motivic cohomology. However, Voevodsky also shows that (with integral coefficients) it does not admit a motivic t-structure. • Start with the category Sm of smooth varieties over a perfect field. Similarly to the construction of pure motives above, instead of usual morphisms smooth correspondences are allowed. Compared to the (quite general) cycles used above, the definition of these correspondences is more restrictive; in particular they always intersect properly, so no moving of cycles and hence no equivalence relation is needed to get a well-defined composition of correspondences. This category is denoted SmCor, it is additive. • As a technical intermediate step, take the bounded homotopy category K (SmCor) of complexes of smooth schemes and correspondences. • Apply localization of categories to force any variety X to be isomorphic to X x A (product with the affine line) and also, that a Mayer- Vietoris-sequence holds, i.e. X= U <u V (union of two open subvarieties) shall be isomorphic to U n V — > U u V. • Finally, as above, take the pseudo-abelian envelope. The resulting category is called the category of effective geometric motives. Again, formally inverting the Tate object, one gets the category DM of geometric motives. Explanation for non-specialists A commonly applied technique in mathematics is to study objects carrying a particular structure by introducing a category whose morphisms preserve this structure. Then one may ask, when are two given objects isomorphic and ask for a "particularly nice" representative in each isomorphism class. The classification of algebraic varieties, i.e. application of this idea in the case of algebraic varieties, is very difficult due to the highly non-linear structure of the objects. The relaxed question of studying varieties up to birational isomorphism has led to the field of birational geometry. Another way to handle the question is to attach to a given variety X an object of more linear nature, i.e. an object amenable to the techniques of linear algebra, for example a vector space. This "linearization" goes usually under the name of cohomology. Motive (algebraic geometry) 201 There are several important cohomology theories which reflect different structural aspects of varieties. The (partly conjectural) theory of motives is an attempt to find a universal way to linearize algebraic varieties, i.e. motives are supposed to provide a cohomology theory which embodies all these particular cohomologies. For example, the genus of a smooth projective curve C which is an interesting invariant of the curve, is an integer, which can be read off the dimension of the first Betti cohomology group of C. So, the motive of the curve should contain the genus information. Of course, the genus is a rather coarse invariant, so the motive of C is more than just this number. The search for a universal cohomology Each algebraic variety X has a corresponding motive [X], so the simplest examples of motives are: • [point] • [projective line] = [point] + [line] • [projective plane] = [plane] + [line] + [point] These 'equations' hold in many situations, namely for de Rham cohomology and Betti cohomology, /-adic cohomology, the number of points over any finite field, and in multiplicative notation for local zeta-functions. The general idea is that one motive has the same structure in any reasonable cohomology theory with good formal properties; in particular, any Weil cohomology theory will have such properties. There are different Weil cohomology theories, they apply in different situations and have values in different categories, and reflect different structural aspects of the variety in question: • Betti cohomology is defined for varieties over (subfields of) the complex numbers, it has the advantage of being defined over the integers and is a topological invariant • de Rham cohomology (for varieties over D) comes with a mixed Hodge structure, it is a differential-geometric invariant • /-adic cohomology (over any field of characteristic * 1) has a canonical Galois group action, i.e. has values in representations of the (absolute) Galois group • crystalline cohomology All these cohomology theories share common properties, e.g. existence of Mayer- Vietoris-sequences, homotopy invariance (H (X)=H (X x A ), the product of X with the affine line) and others. Moreover, they are linked by comparison isomorphisms, for example Betti cohomology H . (X, Win) of a smooth variety X over D with finite coefficients is isomorphic to /-adic cohomology with finite coefficients. The theory of motives is an attempt to find a universal theory which embodies all these particular cohomologies and their structures and provides a framework for "equations" like [projective line] = [line] + [point]. In particular, calculating the motive of any variety X directly gives all the information about the several Weil cohomology theories H„ . (X), H„„ (X) etc. ° J Betti J ' DR v ' Beginning with Grothendieck, people have tried to precisely define this theory for many years. Motive (algebraic geometry) 202 Motivic cohomology Motivic cohomology itself had been invented before the creation of mixed motives by means of algebraic K-theory. The above category provides a neat way to (re)define it by H'\X, m) := H n (X, D(m)) := Hom DM (X, D(m)[«]), where n and m are integers and D(m) is the m-th tensor power of the Tate object 0(1), which in Voevodsky's setting is the complex D — > point shifted by -2, and [«] means the usual shift in the triangulated category. Conjectures related to motives The standard conjectures were first formulated in terms of the interplay of algebraic cycles and Weil cohomology theories. The category of pure motives provides a categorical framework for these conjectures. The standard conjectures are commonly considered to be very hard and are open in the general case. Grothendieck, with Bombieri, showed the depth of the motivic approach by producing a conditional (very short and elegant) proof of the Weil conjectures (which are proven by different means by Deligne), assuming the standard conjectures to hold. For example, the Kiinneth standard conjecture, which states the existence of algebraic cycles n C X x X inducing the canonical projectors H (X) H l (X) H (X) (for any Weil cohomology H) implies that every pure motive M decomposes in graded pieces of weight n: M = ®Gr M. The terminology weights comes from a similar decomposition of, say, de-Rham cohomology of smooth projective varieties, see Hodge theory. Conjecture D, stating the concordance of numerical and homological equivalence, implies the equivalence of pure motives with respect to homological and numerical equivalence. (In particular the former category of motives would not depend on the choice of the Weil cohomology theory). Jannsen (1992) proved the following unconditional result: the category of (pure) motives over a field is abelian and semisimple if and only if the chosen equivalence relation is numerical equivalence. The Hodge conjecture, may be neatly reformulated using motives: it holds iff the Hodge realization mapping any pure motive with rational coefficients (over a subfield k of D) to its Hodge structure is a full functor H : M(fc)n — > HSn (rational Hodge structures). Here pure motive means pure motive with respect to homological equivalence. Similarly, the Tate conjecture is equivalent to: the so-called Tate realization, i.e. ^-adic cohomology is a faithful functor H : M(k)n£ — > Rep (Gal(k)) (pure motives up to homological equivalence, continuous representations of the absolute Galois group of the base field k), which takes values in semi-simple representations. (The latter part is automatic in the case of the Hodge analogue). Tannakian formalism and motivic Galois group To motivate the (conjectural) motivic Galois group, fix a field k and consider the functor finite separable extensions Kofk -^finite sets with a (continuous) action of the absolute Galois group ofk which maps K to the (finite) set of embeddings of K into an algebraic closure of k. In Galois theory this functor is shown to be an equivalence of categories. Notice that fields are O-dimensional. Motives of this kind are called Artin motives. By D-linearizing the above objects, another way of expressing the above is to say that Artin motives are equivalent to finite D-vector spaces together with an action of the Galois group. The objective of the motivic Galois group is to extend the above equivalence to higher-dimensional varieties. In order to do this, the technical machinery of Tannakian category theory (going back to Tannaka-Krein duality, but a purely algebraic theory) is used. Its purpose is to shed light on both the Hodge conjecture and the Tate conjecture, the outstanding questions in algebraic cycle theory. Fix a Weil cohomology theory H. It gives a functor from M YiiAYYi (pure motives using numerical equivalence) to finite-dimensional D-vector spaces. It can be shown that the former category is a Tannakian category. Assuming the equivalence of homological and numerical equivalence, i.e. the Motive (algebraic geometry) 203 above standard conjecture D, the functor H is an exact faithful tensor-functor. Applying the Tannakian formalism, one concludes that M is equivalent to the category of representations of an algebraic group G, which is called motivic Galois group. It is to the theory of motives what the Mumford-Tate group is to Hodge theory. Again speaking in rough terms, the Hodge and Tate conjectures are types of invariant theory (the spaces that are morally the algebraic cycles are picked out by invariance under a group, if one sets up the correct definitions). The motivic Galois group has the surrounding representation theory. (What it is not, is a Galois group; however in terms of the Tate conjecture and Galois representations on etale cohomology, it predicts the image of the Galois group, or, more accurately, its Lie algebra.) References • Andre, Yves (2004), Une introduction aux motifs (motifs purs, motifs mixtes, periodes), Panoramas et Syntheses, 17, Paris: Societe Mathematique de France, MR21 15000, ISBN 978-2-85629-164-1 • Beilinson, Alexander; Vologodsky, Vadim (2007), A guide to Voevodsky's motives (technical introduction with comparatively short proofs) • Jannsen, Uwe (1992), "Motives, numerical equivalence and semi-simplicity", Inventions math. 107: 447—452, doi:10.1007/BF01231898 • Uwe Jannsen ... eds. (1994), Jannsen, Uwe; Kleiman, Steven; Serre, Jean-Pierre, eds., Motives, Proceedings of Symposia in Pure Mathematics, 55, Providence, R.I.: American Mathematical Society, MR1265518, ISBN 978-0-8218-1636-3 • L. Breen: Tannakian categories. • S. Kleiman: The standard conjectures. • A. Scholl: Classical motives, (detailed exposition of Chow motives) • Kleiman, Steven L. (1972), "Motives", in Oort, F., Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer-School in Math., Oslo, 1970), Groningen: Wolters-Noordhoff, pp. 53—82 (adequate equivalence relations on cycles). Hi • Mazur, Barry (2004), "What is ... a motive?" , Notices of the American Mathematical Society 51 (10): 1214-1216, MR2104916, ISSN 0002-9920 (motives-for-dummies text). • Mazza, Carlo; Voevodsky, Vladimir; Weibel, Charles (2006), Lecture notes on motivic cohomology , Clay Mathematics Monographs, 2, Providence, R.I.: American Mathematical Society, MR2242284, ISBN 978-0-8218-3847-1; 978-0-8218-3847-1 • Milne, James S. Motives — Grothendieck's Dream • Serre, Jean-Pierre (1991), "Motifs", Asterisque (198): 11, 333-349 (1992), MR1 144336, ISSN 0303-1179 (non-technical introduction to motives). • Voevodsky, Vladimir; Suslin, Andrei; Friedlander, Eric M. (2000), Cycles, transfers, and motivic homology theories [ , Annals of Mathematics Studies, Princeton, NJ: Princeton University Press, ISBN 978-0-691-04814-7; 978-0-691-04815-4 (Voevodsky's definition of mixed motives. Highly technical). Motive (algebraic geometry) 204 References [ 1 ] http : // w w w . math. uiuc. edu/K- theory/08 3 2/ [2] http://www.ams.org/notices/200410/what-is.pdf [3] http://math.rutgers.edu/~weibel/motiviclectures.html [4] http://www.jmilne.org/math/xnotes/MOT.pdf [5] http://www.math.uiuc.edu/K-theory/0368/ Grothendieck— Hirzebruch— Riemann— Roch theorem In mathematics, specifically in algebraic geometry, the Grothendieck— Riemann— Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch— Riemann— Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann— Roch theorem for line bundles on compact Riemann surfaces. Riemann— Roch type theorems relate Euler characteristics of the cohomology of a vector bundle with their topological degrees, or more generally their characteristic classes in (co)homology or algebraic analogues thereof. The classical Riemann— Roch theorem does this for curves and line bundles, whereas the Hirzebruch— Riemann— Roch theorem generalises this to vector bundles over manifolds. The Grothendieck— Hirzebruch— Riemann— Roch theorem sets both theorems in a relative situation of a morphism between two manifolds (or more general schemes) and changes the theorem from a statement about a single bundle, to one applying to chain complexes of sheaves. The theorem has been very influential, not least for the development of the Atiyah— Singer index theorem. Conversely, complex analytic analogues of the Grothendieck— Hirzebruch— Riemann— Roch theorem can be proved using the families index theorem. Alexander Grothendieck, its author, was rumored to have finished the proof around 1956 but did not publish his theorem because he was not satisfied with it. Instead Armand Borel and Jean-Pierre Serre wrote up and published Grothendieck' s preliminary (as he saw it) proof. Formulation Let X be a smooth quasi-projective scheme over a field. Under these assumptions, the Grothendieck group K {X) of bounded complexes of coherent sheaves is canonically isomorphic to the Grothendieck group of bounded complexes of finite-rank vector bundles. Using this isomorphism, consider the Chern character (a rational combination of Chern classes) as a functorial transformation ch: K (X) -> A(X,Q) where A d (X,Q) is the Chow group of cycles on X of dimension d modulo rational equivalence, tensored with the rational numbers. In case X is defined over the complex numbers, the latter group maps to the topological cohomology group # 2dim W- 2rf (X,Q). Now consider a proper morphism f-.X^Y between smooth quasi-projective schemes and a bounded complex of sheaves jP" B . The Grothendieck— Riemann— Roch theorem relates the push forward maps /, = £(-! W*: K (X) -► K (Y) Grothendieck— Hirzebruch— Riemann— Roch theorem 205 and the pushforward f.:A(X)^A(Y), by the formula ch(/,jp--)td(y) = /.(ch(jp-)td(x)). Here td(X) is the Todd genus of (the tangent bundle of) X. Thus the theorem gives a precise measure for the lack of commutativity of taking the push forwards in the above senses and the Chern character and shows that the needed correction factors depends on X and Y only. In fact, since the Todd genus is functorial and multiplicative in exact sequences, we can rewrite the Grothendieck Hirzebruch Riemann Roch formula to cli(/,^-) = /,(ch(^-)td(T / )) where Tjis the relative tangent sheaf of /. This is often useful in applications, for example if/ is a locally trivial fibration. Generalising and specialising Generalisations of the theorem can be made to the non-smooth case by considering an appropriate generalisation of the combination ch( — )td(X) and to the non-proper case by considering cohomology with compact support. The arithmetic Riemann— Roch theorem extends the Grothendieck— Riemann— Roch theorem to arithmetic schemes. The Hirzebruch— Riemann— Roch theorem is (essentially) the special case where Y is a point and the field is the field of complex numbers. History Grothendieck's version of the Riemann— Roch theorem was originally conveyed in a letter to Serre around 1956—7. It was made public at the initial Bonn Arbeitstagung, in 1957. Serre and Armand Borel subsequently organized a seminar at Princeton to understand it. The final published paper was in effect the Borel— Serre exposition. The significance of Grothendieck's approach rests on several points. First, Grothendieck changed the statement itself: the theorem was, at the time, understood to be a theorem about a variety, whereas after Grothendieck, it was known to essentially be understood as a theorem about a morphism between varieties. In short, he applied a strong categorical approach to a hard piece of analysis. Moreover, Grothendieck introduced K-groups, as discussed above, which paved the way for algebraic K theory. References • Borel, Armand; Serre, Jean-Pierre (1958), "Le theoreme de Riemann— Roch" , Bulletin de la Societe Mathematique de France 86: 97-136, MR01 16022, ISSN 0037-9484 • Fulton, William (1998), Intersection theory, Berlin, New York: Springer-Verlag, MR1644323, ISBN 978-3-540-62046-4; 978-0-387-98549-7 References [1] http://www.numdam.org/item?id=BSMF_1958_86_97_0 Coherent sheaf 206 Coherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometrical information. In addition, there is a related concept of quasi-coherent sheaves. Many results and properties in algebraic geometry and complex analytic geometry are formulated in terms of coherent sheaves and their cohomology. Coherent sheaves can be seen as a generalization of (sheaves of sections of) vector bundles. They form a category closed under usual operations such as taking kernels, cokernels and finite direct sums. In addition, under suitable compactness conditions they are preserved under maps of the underlying spaces and have finite dimensional cohomology spaces. Definition A coherent sheaf on a ringed space (X, Ox) l& a sheaf ^"of Ox -modules with the following two properties: 1. ^"is of finite type over Ox > i- e -> f° r anv point x €E X there is an open neighbourhood JJ (2 X such that the restriction J?\ ^of J? to JJ is generated by a finite number of sections (in other words, there is a surjective morphism 0%\\j — ► J-\ufor some fi (z fsj); and 2. for any open set JJ (2 X > anv n £ N an d a ny morphism : 0% I u — > J-\ [/of Ox -modules, the kernel of (j) is of finite type. The sheaf of rings Ox * s coherent if it is coherent considered as a sheaf of modules over itself. Important examples of coherent sheaves of rings include the sheaf of germs of holomorphic functions on a complex manifolds and the structure sheaf of a Noetherian scheme from algebraic geometry. A coherent sheaf is always a sheaf of finite presentation, or in other words each point x €E X nas an open neighbourhood {/such that the restriction jFlyOf ^to [/is isomorphic to the cokernel of a morphism ®x I U — > Ov I u for some integers n and m . If Ox l& coherent, then the converse is true and each sheaf of finite presentation over Ox l& coherent. For a sheaf of rings O > a sheaf J^oi O -modules is said to be quasi-coherent if it has a local presentation, i.e. if there exist an open cover by Ui of the topological space and an exact sequence i *-*i i ^i i ^i where the first two terms of the sequence are direct sums (possibly infinite) of copies of the structure sheaf. For an affine variety X with (affine) coordinate ring R, there exists a covariant equivalence of categories between that of quasi-coherent sheaves and sheaf morphisms on the one hand, and R-modules and module homomorphisms on the other hand. In case the ring R is Noetherian, coherent sheaves correspond exactly to finitely generated modules. Coherence of sheaves is working in the background of some results in commutative algebra, e.g. Nakayama's lemma, which in terms of sheaves says that if J^is a coherent sheaf, then the fiber k(x) ®o x 3~x = ^if an d only if there is a neighborhood JJ of x so that J-\jj = 0- The role played by coherent sheaves is as a class of sheaves, say on an algebraic variety or complex manifold, that is more general than the locally free sheaf — such as invertible sheaf, or sheaf of sections of a (holomorphic) vector bundle — but still with manageable properties. The generality is desirable, to be able to take kernels and cokernels of morphisms, for example, without moving outside the given class of sheaves. Coherent sheaf 207 Examples of coherent sheaves • On noetherian schemes, the structure sheaf O itself. • Sheaves of sections in vector bundles. • The Oka coherence theorem shows that the sheaf of holomorphic functions on a complex manifold is coherent. • Ideal sheaves: If Z is a closed complex subspace of a complex analytic space X, the sheaf / of all holomorphic functions vanishing on Z is coherent. • Structure sheaves of subspaces. Coherent cohomology The sheaf cohomology theory of coherent sheaves is called coherent cohomology. It is one of the major and most fruitful applications of sheaves, and its results connect quickly with classical theories. Using a theorem of Schwartz on compact operators in Frechet spaces, Cartan and Serre proved that compact complex manifolds have the property that their sheaf cohomology for any coherent sheaf consists of vector spaces of finite dimension. This result had been proved previously by Kodaira for the particular case of locally free sheaves on Kahler manifolds. It plays a major role in the proof of the "GAGA" equivalence analytic <-> algebraic. An algebraic (and much easier) version of this theorem was proved by Serre. Relative versions of this result for a proper morphism were proved by Grothendieck in the algebraic case and by Grauert and Remmert in the analytic case. For example Grothendieck' s result concerns the functor R/^ or push-forward, in sheaf cohomology. (It is the right derived functor of the direct image of a sheaf.) For a proper morphism in the sense of scheme theory, it was shown that this functor sends coherent sheaves to coherent sheaves. The result of Serre is the case of a morphism to a point. The duality theory in scheme theory that extends Serre duality is called coherent duality (or Grothendieck duality). Under some mild conditions of finiteness, the sheaf of Kahler differentials on an algebraic variety is a coherent sheaf Q. . When the variety is non-singular its 'top' exterior power acts as the dualising object; and it is locally free (effectively it is the sheaf of sections of the cotangent bundle, when working over the complex numbers, but that is a statement that requires more precision since only holomorphic 1 -forms count as sections). The successful extension of the theory beyond this case was a major step. References Danilov, V. I. (2001), "Coherent algebraic sheaf" , in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104 • Section 0.5.3 of Grothendieck, Alexandre; Dieudonne, Jean (1960). "Elements de geometrie algebrique (rediges avec la collaboration de Jean Dieudonne) : I. Le langage des schemas" . Publications Mathematiques de 1'IHES 4. MR0217083. • Robin Hartshorne, Algebraic Geometry, Springer- Verlag, 1977, ISBN 0-387-90244-9 • Onishchik, A.L. (2001), "Coherent analytic sheaf" , in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104 Onishchik, A.L. (2001), ISBN 978-1556080104 Onishchik, A.L. (2001), "Coherent sheaf" , in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, Coherent sheaf 208 References [1] http://eom. springer. de/c/c022980.htm [2] http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1960 4_ [3] http://eom. springer. de/c/c022990.htm [4] http://eom.springer.de/C/c023020.htm Grothendieck topology In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site. Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the etale cohomology of a scheme. It has been used to define other cohomology theories since then, such as 1-adic cohomology, flat cohomology, and crystalline cohomology. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry. There is a natural way to associate a site to an ordinary topological space, and Grothendieck' s theory is loosely regarded as a generalization of classical topology. Under meager point-set hypotheses, namely sobriety, this is completely accurate — it is possible to recover a sober space from its associated site. However simple examples such as the indiscrete topological space show that not all topological spaces can be expressed using Grothendieck topologies. Conversely, there are Grothendieck topologies which do not come from topological spaces. Introduction Andre Weil's famous Weil conjectures proposed that certain properties of equations with integral coefficients should be understood as geometric properties of the algebraic variety that they defined. His conjectures postulated that there should be a cohomology theory of algebraic varieties which gave number-theoretic information about their defining equations. This cohomology theory was known as the "Weil cohomology", but using the tools he had available, Weil was unable to construct it. In the early 1960s, Alexander Grothendieck introduced etale maps into algebraic geometry as algebraic analogues of local analytic isomorphisms in analytic geometry. He used etale coverings to define an algebraic analogue of the fundamental group of a topological space. Soon Jean-Pierre Serre noticed that some properties of etale coverings mimicked those of open immersions, and that consequently it was possible to make constructions which imitated the cohomology functor H . Grothendieck saw that it would be possible to use Serre's idea to define a cohomology theory which he suspected would be the Weil cohomology. To define this cohomology theory, Grothendieck needed to replace the usual, topological notion of an open covering with one that would use etale coverings instead. Grothendieck also saw how to phrase the definition of covering abstractly; this is where the definition of a Grothendieck topology comes from. Grothendieck topology 209 Definition Motivation The classical definition of a sheaf begins with a topological space X. A sheaf associates information to the open sets of X. This information can be phrased abstractly by letting 0{X) be the category whose objects are the open subsets U of X and whose morphisms are the inclusion maps V — > U of open sets U and V of X. We will call such maps open immersions, just as in the context of schemes. Then a presheaf on X is a contravariant functor from 0{X) to the category of sets, and a sheaf is a presheaf which satisfies the gluing axiom. The gluing axiom is phrased in terms of pointwise covering, i.e., {[/.} covers U if and only if U . U. = U. In this definition, U. is an open subset of X. Grothendieck topologies replace each U. with an entire family of open subsets; in this example, U. is replaced by the family of all open immersions V.. — > U .. Such a collection is called a sieve. Pointwise covering is replaced by the notion of a covering family; in the above example, the set of all {V.. — > U .} . as i varies is a covering family of U. Sieves and covering families can be axiomatized, and once this is done open sets and pointwise covering can be replaced by other notions which describe other properties of the space X. Sieves In a Grothendieck topology, the notion of a collection of open subsets of U stable under inclusion is replaced by the notion of a sieve. If c is any given object in C, a sieve on c is a subfunctor of the functor Hom(-, c); (this is the Yoneda embedding applied to c). In the case of 0(X), a sieve S on an open set U selects a collection of open subsets of U which is stable under inclusion. More precisely, consider that for any open subset V of U, S(V) will be a subset of Hom( V, U), which has only one element, the open immersion V — > U. Then V will be considered "selected" by S if and only if S(V) is nonempty. If W is a subset of V, then there is a morphism S(V) — > S(W) given by composition with the inclusion W — > V. If S(V) is non-empty, it follows that S(W) is also non-empty. If S is a sieve on X, and/: Y — > X is a morphism, then left composition by /gives a sieve on Y called the pullback of S along / denoted by / * S. It is defined as the fibered product S x Hom(-, Y) together with its natural embedding in Hom(-, Y). More concretely, for each object Z of C, / *S{Z) = { g: Z — » Y I fg £ S(Z) }, and/ *5 inherits its action on morphisms by being a subfunctor of Hom(-, Y). In the classical example, the pullback of a collection { V. } of subsets of U along an inclusion W —> U is the collection { V.nW } . Covering Sieves A classical topology on a set X is a collection of distinguished subsets, called open sets. This selection is subject to certain conditions: the axioms of a topological space. By comparison, a Grothendieck topology J on a category C is a collection, for each object c of C, of distinguished sieves on c, called covering sieves of c and denoted by J{c). This selection will be subject to certain axioms, stated below. Continuing the previous example, a sieve S on an open set U in 0{X) will be a covering sieve if and only if the union of all the open sets V for which S(V) is nonempty equals U; in other words, if and only if S gives us a collection of open sets which cover U in the classical sense. Axioms The conditions we impose on a Grothendieck topology are: • (T 1) (Base change) If S is a covering sieve on X, and/ Y — » X is a morphism, then the pullback/ * S is a covering sieve on 7. • (T 2) (Local character) Let S be a covering sieve on X, and let Tbe any sieve on X Suppose that for each object Y of C and each arrow/ Y — > X in S^F), the pullback sieve/ * T is a covering sieve on Y. Then T is a covering sieve on! • (T 3) (Identity) Hom(-, X) is a covering sieve on X for any object X in C Grothendieck topology 210 The base change axiom corresponds to the idea that if { Ui } covers U, then [U. n V} should cover U n V. The local character axiom corresponds to the idea that if { U 1 covers U and { V } £ i covers U for each 2, then the i u J J ' collection {V..} for all i and j should cover U. Lastly, the identity axiom corresponds to the idea that any set is covered by all its possible subsets. Alternative Axioms In fact, it is possible to put these axioms in another form where their geometric character is more apparent, assuming that the underlying category C contains certain fibered products. In this case, instead of specifying sieves, we can specify that certain collections of maps with a common codomain should cover their codomain. These collections are called covering families. If the collection of all covering families satisfies certain axioms, then we say that they form a Grothendieck pretopology. These axioms are: • (PT 0) (Existence of fibered products) For all objects X of C, and for all morphisms X — > X which appear in some covering family of X, and for all morphisms 7— > X, the fibered product X x Y exists. a • (PT 1) (Stability under base change) For all objects X of C, all morphisms Y — > X, and all covering families {X — » X], the family {X x Y — > Y] is a covering family. • (PT 2) (Local character) If {X — > X] is a covering family, and if for all a, {X — » X } is a covering family, then the family of composites {X — > X — > X] is a covering family. • (PT 3) (Isomorphisms) Iff. Y — » X is an isomorphism, then {/) is a covering family. For any pretopology, the collection of all sieves that contain a covering family from the pretopology is always a Grothendieck topology. For categories with fibered products, there is a converse. Given a collection of arrows {X — > X], we construct a sieve S by letting S(Y) be the set of all morphisms Y — > X that factor through some arrow X — > X. This is called the sieve generated by {X — > X}. Now choose a topology. Say that {X — > X] is a covering family if and only if the sieve that it generates is a covering sieve for the given topology. It is easy to check that this defines a pretopology. (PT 3) is sometimes replaced by a weaker axiom: • (PT 3') (Identity) If 1 : X — > X is the identity arrow, then {7 } is a covering family. (PT 3) implies (PT 3'), but not conversely. However, suppose that we have a collection of covering families that satisfies (PT 0) through (PT 2) and (PT 3'), but not (PT 3). These families generate a pretopology. The topology generated by the original collection of covering families is then the same as the topology generated by the pretopology, because the sieve generated by an isomorphism Y — > X is Hom(-, X). Consequently, if we restrict our attention to topologies, (PT 3) and (PT 3') are equivalent. Sites and sheaves Let C be a category and let / be a Grothendieck topology on C. The pair (C, J) is called a site. A presheaf on a category is a contravariant functor from C to the category of all sets. Note that for this definition C is not required to have a topology. A sheaf on a site, however, should allow gluing, just like sheaves in classical topology. Consequently, we define a sheaf on a site to be a presheaf F such that for all objects X and all covering sieves S on X, the natural map Hom(Hom(-, X), F) — > Hom(5', F) induced by the inclusion of S into Hom(-, X) is a bijection. Halfway in between a presheaf and a sheaf is the notion of a separated presheaf, where the natural map above is required to be only an injection, not a bijection, for all sieves S. A morphism of presheaves or of sheaves is a natural transformation of functors. The category of all sheaves on C is the topos defined by the site (C, J). Using the Yoneda lemma, it is possible to show that a presheaf on the category 0(X) is a sheaf on the topology defined above if and only if it is a sheaf in the classical sense. Sheaves on a pretopology have a particularly simple description: For each covering family {X — » X], the diagram Grothendieck topology 211 f(x) -> n F(x a ) zz n f ( x « x * x ?) aeA a,@eA must be an equalizer. For a separated presheaf, the first arrow need only be injective. Similarly, one can define presheaves and sheaves of abelian groups, rings, modules, and so on. One can require either that a presheaf F is a contravariant functor to the category of abelian groups (or rings, or modules, etc.), or that F be an abelian group (ring, module, etc.) object in the category of all contravariant functors from C to the category of sets. These two definitions are equivalent. Examples of sites The discrete and indiscrete topologies Let C be any category. To define the discrete topology, we declare all sieves to be covering sieves. If C has all fibered products, this is equivalent to declaring all families to be covering families. To define the indiscrete topology, we declare only the sieves of the form Hom(-, X) to be covering sieves. The indiscrete topology is also known as the biggest or chaotic topology, and it is generated by the pretopology which has only isomorphisms for covering families. A sheaf on the indiscrete site is the same thing as a presheaf. The canonical topology Let C be any category. The Yoneda embedding gives a functor Hom(-, X) for each object X of C. The canonical topology is the biggest topology such that every representable presheaf Hom(-, X) is a sheaf. A covering sieve or covering family for this site is said to be strictly universally epimorphic. A topology which is less fine than the canonical topology, that is, for which every covering sieve is strictly universally epimorphic, is called subcanonical. Subcanonical sites are exactly the sites for which every presheaf of the form Hom(-, X) is a sheaf. Most sites encountered in practice are subcanonical. Small site associated to a topological space We repeat the example which we began with above. Let X be a topological space. We defined 0{X) to be the category whose objects are the open sets of X and whose morphisms are inclusions of open sets. The covering sieves on an object U of 0(X) were those sieves S satisfying the following condition: • If W is the union of all the sets V such that S(V) is non-empty, then W-U. This topology can also naturally be expressed as a pretopology. We say that a family of inclusions {V C U] is a covering family if and only if the union U V equals U. This site is called the small site associated to a topological space X. Big site associated to a topological space Let Spc be the category of all topological spaces. Given any family of functions {u : V — > X], we say that it is a surjective family or that the morphisms u are jointly surjective if (J u (V ) equals X. We define a pretopology on Spc by taking the covering families to be surjective families all of whose members are open immersions. Let S be a sieve on Spc. S is a covering sieve for this topology if and only if: • For all Y and every morphism/ : Y — > X in S(Y), there exists a V and a g : V — > X such that g is an open immersion, g is in S(V), and/factors through g. • If W is the union of all the sets /(F), where/ : Y — > X is in S(Y), then W = X. Fix a topological space X. Consider the comma category Spc/X of topological spaces with a fixed continuous map to X. The topology on Spc induces a topology on Spc/X. The covering sieves and covering families are almost exactly the same; the only difference is that now all the maps involved commute with the fixed maps to X. This is the big Grothendieck topology 212 site associated to a topological space X . Notice that Spc is the big site associated to the one point space. This site was first considered by Jean Giraud. The big and small sites of a manifold Let M be a manifold. M has a category of open sets 0{M) because it is a topological space, and it gets a topology as in the above example. For two open sets U and V of M, the fiber product Ux Vis the open set U n V, which is still in 0(M). This means that the topology on 0{M) is defined by a pretopology, the same pretopology as before. Let Mfd be the category of all manifolds and continuous maps. (Or smooth manifolds and smooth maps, or real analytic manifolds and analytic maps, etc.) Mfd is a subcategory of Spc, and open immersions are continuous (or smooth, or analytic, etc.), so Mfd inherits a topology from Spc. This lets us construct the big site of the manifold M as the site Mfd/M. We can also define this topology using the same pretopology we used above. Notice that to satisfy (PT 0), we need to check that for any continuous map of manifolds X — > Y and any open subset U of Y, the fibered product U x X is in Mfd/M. This is just the statement that the preimage of an open set is open. Notice, however, that not all fibered products exist in Mfd because the preimage of a smooth map at a critical value need not be a manifold. Topologies on the category of schemes The category of schemes, denoted Sch, has a tremendous number of useful topologies. A complete understanding of some questions may require examining a scheme using several different topologies. All of these topologies have associated small and big sites. The big site is formed by taking the entire category of schemes and their morphisms, together with the covering sieves specified by the topology. The small site over a given scheme is formed by only taking the objects and morphisms which are part of a cover of the given scheme. The most elementary of these is the Zariski topology. Let X be a scheme. X has an underlying topological space, and this topological space determines a Grothendieck topology. The Zariski topology on Sch is generated by the pretopology whose covering families are jointly surjective families of scheme-theoretic open immersions. The covering sieves S for Zar are characterized by the following two properties: • For all Y and every morphism/ : Y — > X in S(Y), there exists a V and a g : V — > X such that g is an open immersion, g is in S(V), and/factors through g. • If W is the union of all the sets f(Y), where/ : Y — > X is in S(Y), then W = X. Despite their outward similarities, the topology on Zar is not the restriction of the topology on Spcl This is because there are morphisms of schemes which are topologically open immersions but which are not scheme-theoretic open immersions. For example, let A be a non-reduced ring and let N be its ideal of nilpotents. The quotient map A — » A/N induces a map Spec A/N — > Spec A which is the identity on underlying topological spaces. To be a scheme-theoretic open immersion it must also induce an isomorphism on structure sheaves, which this map does not do. In fact, this map is a closed immersion. The etale topology is finer than the Zariski topology. It was the first Grothendieck topology to be closely studied. Its covering families are jointly surjective families of etale morphisms. It is finer than the Nisnevich topology, but neither finer nor coarser than the cdh and 1' topologies. There are two flat topologies, the fppf topology and the fpqc topology. In the fppf topology, covering morphisms are finitely presented and faithfully flat, and in the fpqc topology, covering morphisms are finitely presented and quasi-compact. These topologies are closely related to descent. The fpqc topology is finer than all the topologies mentioned above, and it is very close to the canonical topology. Grothendieck introduced crystalline cohomology to study the p-torsion part of the cohomology of characteristic p varieties. In the crystalline topology which is the basis of this theory, covering maps are given by infinitesimal thickenings together with divided power structures. The crystalline covers of a fixed scheme form a category with no final object. Grothendieck topology 213 Continuous and cocontinuous functors There are two natural types of functors between sites. They are given by functors which are compatible with the topology in a certain sense. Continuous functors If (C, J) and (D, K) are sites and u : C — > D is a functor, then u is continuous if for every sheaf F on D with respect to the topology K, the presheaf Fu is a sheaf with respect to the topology J. Continuous functors induce functors between the corresponding topoi by sending a sheaf F to Fu. These functors are called pushforwards. If q and f\ denote the topoi associated to C and D, then the pushforward functor is u ■ f) j. Q . u admits a left adjoint u called the pullback. u need not preserve limits, even finite limits. In the same way, u sends a sieve on an object X of C to a sieve on the object uX of D. A continuous functor sends covering sieves to covering sieves. If J is the topology defined by a pretopology, and if u commutes with fibered products, then u is continuous if and only if it sends covering sieves to covering sieves and if and only if it sends covering families to covering families. In general, it is not sufficient for u to send covering sieves to covering sieves (see SGA IV 3, Exemple 1.9.3). Cocontinuous functors Again, let (C, J) and (D, K) be sites and v : C — > D be a functor. If X is an object of C and R is a sieve on vX, then R can be pulled back to a sieve S as follows: A morphism/ : Z — > X is in S if and only if v(f) : vZ — > vX is in R. This defines a sieve, v is cocontinuous if and only if for every object X of C and every covering sieve R of vX, the pullback S of 7? is a covering sieve on X. Composition with v sends a presheaf F on D to a presheaf Fv on C, but if v is cocontinuous, this need not send sheaves to sheaves. However, this functor on presheaf categories, usually denoted y* , admits a right adjoint v^ . Then v is cocontinuous if and only if v^ sends sheaves to sheaves, that is, if and only if it restricts to a functor v ■ Q j. £) . In this case, the composite of {j* with the associated sheaf functor is a left adjoint of v^ denoted v . Furthermore, v preserves finite limits, so the adjoint functors v„. and v determine a geometric morphism of topoi c -^b- Morphisms of sites A continuous functor u : C — > D is a morphism of sites D — > C («of C — > D) if m preserves finite limits. In this case, u and u determine a geometric morphism of topoi (j •, f) ■ The reasoning behind the convention that a continuous functor C — > D is said to determine a morphism of sites in the opposite direction is that this agrees with the intuition coming from the case of topological spaces. A continuous map of topological spaces X — > Y determines a continuous functor 0{Y) — > 0{X). Since the original map on topological spaces is said to send X to Y, the morphism of sites is said to as well. A particular case of this happens when a continuous functor admits a left adjoint. Suppose that u : C — > D and v : D — > C are functors with u right adjoint to v. Then u is continuous if and only if v is cocontinuous, and when this happens, u is naturally isomorphic to v and u is naturally isomorphic to v„.. In particular, u is a morphism of sites. Grothendieck topology 214 Bibliography • Artin, Michael (1962), Grothendieck topologies, Harvard University, Dept. of Mathematics • Demazure, Michel; Alexandre Grothendieck, eds. (1970) (in French). Seminaire de Geometrie Algebrique du Bois Marie - 1962-64 - Schemas en groupes - (SGA 3) - vol. 1 (Lecture notes in mathematics 151). Berlin; New York: Springer- Verlag. pp. xv+564. • Artin, Michael; Alexandre Grothendieck, Jean-Louis Verdier, eds. (1972) (in French). Seminaire de Geometrie Algebrique du Bois Marie - 1963-64 - Theorie des topos et cohomologie etale des schemas - (SGA 4) - vol. 1 (Lecture notes in mathematics 269). Berlin; New York: Springer- Verlag. xix+525. • Nisnevich, Yevsey A. (1989). "The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory". In J. F. Jardine and V. P. Snaith. Algebraic K-theory: connections with geometry and topology. Proceedings of the NATO Advanced Study Institute held in Lake Louise, Alberta, December 7—11, 1987. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 279. Dordrecht: Kluwer Academic Publishers Group, pp. 241—342., available at Nisnevich's website References [1] http://www.cims.nyu.edu/~nisnevic/ Crystalline cohomology In mathematics, crystalline cohomology is a Weil cohomology theory for schemes introduced by Alexander Grothendieck (1966, 1968) and developed by Pierre Berthelot (1974). Its values are modules over rings of Witt vectors over the base field. Crystalline cohomology is partly inspired by the p-adic proof in Dwork (1960) of part of the Weil conjectures and is closely related to the (algebraic) de Rham cohomology introduced by Grothendieck (1963). Roughly speaking, crystalline cohomology of a variety X in characteristic p is the de Rham cohomology of a smooth lift of X to characteristic 0, while de Rham cohomology of X is the crystalline cohomology reduced mod p (after taking into account higher Tors). The idea of crystalline cohomology, roughly, is to replace the Zariski open sets of a scheme by infinitesimal thickenings of Zariski open sets with divided power structures. The motivation for this is that it can then be calculated by taking a local lifting of a scheme from characteristic p to characteristic and employing an appropriate version of algebraic de Rham cohomology. Crystalline cohomology only works well for smooth proper schemes. Rigid cohomology extends it to more general schemes. Applications For schemes in characteristic p, crystalline cohomology theory can handle questions about p-torsion in cohomology groups better than p-adic etale cohomology. This makes it a natural backdrop for much of the work on p-adic L-functions. Crystalline cohomology, from the point of view of number theory, fills a gap in the 1-adic cohomology information, which occurs exactly where there are 'equal characteristic primes'. Traditionally the preserve of ramification theory, crystalline cohomology converts this situation into Dieudonne module theory, giving an important handle on arithmetic problems. Conjectures with wide scope on making this into formal statements were enunciated by Jean-Marc Fontaine, the resolution of which is called p-adic Hodge theory. Crystalline cohomology 215 de Rham cohomology De Rham cohomology solves the problem of finding an algebraic definition of the cohomology groups (singular cohomology) for X a smooth complex variety. These groups are the cohomology of the complex of smooth differential forms on X (with complex number coefficients), as these form a resolution of the constant sheaf C. The algebraic de Rham cohomology is defined to be the hypercohomology of the complex of algebraic forms (Kahler differentials) on X. The smooth /-forms form an acyclic sheaf, so the hypercohomology of the complex of smooth forms is the same as its cohomology, and the same is true for algebraic sheaves of /-forms over affine varieties, but algebraic sheaves of /-forms over non-affine varieties can have non-vanishing higher cohomology groups, so the hypercohomology can differ from the cohomology of the complex. For smooth complex varieties Grothendieck (1963) showed that the algebraic de Rham cohomology is isomorphic to the usual smooth de Rham cohomology and therefore (by de Rham's theorem) to the cohomology with complex coefficients. This definition of algebraic de Rham cohomology is available for algebraic varieties over any field k. Coefficients If X is a variety over an algebraically closed field of characteristic p > 0, then the 1-adic cohomology groups for / any prime number other than p give satisfactory cohomology groups of X, with coefficients in the ring Z of /-adic integers. It is not possible in general to find similar cohomology groups with coefficients in the p-adic numbers (or the rationals, or the integers). The classic reason (due to Serre) is that if X is a supersingular elliptic curve, then its ring of endomorphisms generates a quaternion algebra over Q that is non-split at p and infinity. If X has a cohomology group over the p-adic integers with the expected dimension 2, the ring of endomorphisms would have a 2-dimensional representation; and this is not possible as it is non-split at p. (A quite subtle point is that if X is a supersingular elliptic curve over the prime field, with p elements, then its crystalline cohomology is a free rank 2 module over the p-adic integers. The argument given does not apply in this case, because some of the endomorphisms of supersingular elliptic curves are only defined over a quadratic extension of the field of order/?.) Grothendieck's crystalline cohomology theory gets around this obstruction because it takes values in the ring of Witt vectors over the ground field. So if the ground field is the algebraic closure of the field of order p, its values are modules over the p-adic completion of the maximal unramified extension of the p-adic integers, a much larger ring containing n-th roots of unity for all n not divisible by p, rather than over the p-adic integers. Motivation One idea for defining a Weil cohomology theory of a variety X over a field k of characteristic p is to 'lift' it to a variety X* over the ring of Witt vectors of k (that gives back X on reduction mod p), then take the de Rham cohomology of this lift. The problem is that it is not at all obvious that this cohomology is independent of the choice of lifting. The idea of crystalline cohomology in characteristic is to find a direct definition of a cohomology theory as the cohomology of constant sheaves on a suitable site Inf(X) over X, called the infinitesimal site and then show it is the same as the de Rham cohomology of any lift. The site Inf(X) is a category whose objects can be thought of as some sort of generalization of the conventional open sets of X. In characteristic its objects are infinitesimal thickenings U— >T of Zariski open subsets U of X. This means that U is the closed subscheme of a scheme T defined by a nilpotent sheaf of ideals on T; for example, Crystalline cohomology 216 Spec(fc)^ Spec(/t[jc]/(x 2 )). Grothendieck showed that for smooth schemes X over C, the cohomology of the sheaf O on Inf(X) is the same as the usual (smooth or algebraic) de Rham cohomology. Crystalline cohomology In characteristic p the most obvious analogue of the crystalline site defined above in characteristic does not work. The reason is roughly that in order to prove exactness of the de Rham complex, one needs some sort of Poincare lemma, whose proof in turn uses integration, and integration requires various divided powers, which exist in characteristic but not always in characteristic p. Grothendieck solved this problem by defining objects of the crystalline site of X to be roughly infinitesimal thickenings of Zariski open subsets of X, together with a divided power structure giving the needed divided powers. We will work over the ring W = W/p n W of Witt vectors of length n over a perfect field k of characteristic p>0. For example, k could be the finite field of order p, and W is then the ring Z/p n Z. (More generally one can work over a base scheme S which has a fixed sheaf of ideals / with a divided power structure.) If X is a scheme over k, then the crystalline site of X relative to W , denoted Cns(X/W ), has as its objects pairs U—>T consisting of a closed immersion of a Zariski open subset U of X into some W -scheme T defined by a sheaf of ideals /, together with a divided power structure on J compatible with the one on W . Crystalline cohomology of a scheme X over k is defined to be the inverse limit IP(X/W) = \imH\XfW n ) where H\XjW n ) = ir{Cris{X/W n ),0) is the cohomology of the crystalline site of X/W with values in the sheaf of rings O = 0„,„ 7 . n X/Wn A key point of the theory is that the crystalline cohomology of a smooth scheme X over k can often be calculated in terms of the algebraic de Rham cohomology of a proper and smooth lifting of X to a scheme Z over W. There is a canonical isomorphism lf(X/W) = H^Z/W) (= lf(Z, Q z/W ) = lirn h\z, n z/Wn )) of the crystalline cohomology of X with the de Rham cohomology of Z over the formal scheme of W (an inverse limit of the hypercohomology of the complexes of differential forms). Conversely the de Rham cohomology of X can be recovered as the reduction mod p of its crystalline cohomology (after taking higher Tors into account). Crystals If X is a scheme over S then the sheaf O is defined by O AT) = coordinate ring of T, where we write T as an A/O X-fo abbreviation for an object £/— »Tof Cris(XAS'). A crystal on the site Cris(X/5') is a sheaf F of O modules that is rigid in the following sense: for any map/between objects T, T of Cns{XIS), the natural map from/ F(T) to F(T) is an isomorphism. This is similar to the definition of a quasicoherent sheaf of modules in the Zariski topology. An example of a crystal is the sheaf O . A/a The term crystal attached to the theory, explained in Grothendieck' s letter to Tate (1966), was a metaphor inspired by certain properties of algebraic differential equations. These had played a role in p-adic cohomology theories (precursors of the crystalline theory, introduced in various forms by Dwork, Monsky, Washnitzer, Lubkin and Katz) particularly in Dwork's work. Such differential equations can be formulated easily enough by means of the algebraic Koszul connections, but in the p-adic theory the analogue of analytic continuation is more mysterious (since p-adic discs tend to be disjoint rather than overlap). By decree, a crystal would have the 'rigidity' and the 'propagation' Crystalline cohomology 217 notable in the case of the analytic continuation of complex analytic functions. (Cf. also the rigid analytic spaces introduced by Tate, in the 1960s, when these matters were actively being debated.) See also • Motivic cohomology • De Rham cohomology References • Berthelot, Pierre (1974), Cohomologie cristalline des schemas de caracteristique p>0, Lecture Notes in Mathematics, Vol. 407, 407, Berlin, New York: Springer- Verlag, doi:10.1007/BFb0068636, MR0384804, ISBN 978-3-540-06852-5 • Berthelot, Pierre; Ogus, Arthur (1978), Notes on crystalline cohomology, Princeton University Press, MR0491705, ISBN 978-0-691-08218-9 • Dwork, Bernard (1960), "On the rationality of the zeta function of an algebraic variety" , American Journal of Mathematics (The Johns Hopkins University Press) 82 (3): 631-648, doi: 10.2307/2372974, MR0140494, ISSN 0002-9327 T21 • Grothendieck, Alexander (1966), "On the de Rham cohomology of algebraic varieties" , Institut des Hautes Etudes Scientifiques. Publications Mathematiques 29 (29): 95-103, doi:10.1007/BF02684807, MR0199194, ISSN 0073-8301 (letter to Atiyah, Oct. 14 1963) • Grothendieck, A. (1966), Letter to J. Tate . mi • Grothendieck, Alexander (1968), "Crystals and the de Rham cohomology of schemes" , in Giraud, Jean; Grothendieck, Alexander; Kleiman, Steven L. et al., Dix Exposes sur la Cohomologie des Schemas, Advanced studies in pure mathematics, 3, Amsterdam: North-Holland, pp. 306—358, MR0269663 • Illusie, Luc (1975), "Report on crystalline cohomology", Algebraic geometry, Proc. Sympos. Pure Math., 29, Providence, R.I.: Amer. Math. Soc, pp. 459-478, MR0393034 • Illusie, Luc (1976), "Cohomologie cristalline (d'apres P. Berthelot)" , Seminaire Bourbaki (1974/1975: Exposes Nos. 453-470), Exp. No. 456, Lecture Notes in Math., 514, Berlin, New York: Springer- Verlag, pp. 53-60, MR0444668 • Illusie, Luc (1994), "Crystalline cohomology", Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., 55, Providence, RI: Amer. Math. Soc, pp. 43-70, MR1265522 References [1] http://www.jstor.org/stable/2372974 [2] http://www.numdam.org/item?id=PMIHES_1966_29_95_0 [3] http://www.math.jussieu.fr/~leila/grothendieckcircle/crystals.pdf [4] http://www.math.jussieu.fr/~leila/grothendieckcircle/DixExp.pdf [5] http://www.numdam.org/numdam-bin/fitem?id=SB_1974-1975 17 53_0 De Rham cohomology 218 De Rham cohomology In mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties. Definition The de Rham complex is the cochain complex of exterior differential forms on some smooth manifold M, with the exterior derivative as the differential. -»• n°(M) 4 n\M) 4 Q 2 (M) 4 fi 3 (M) -► ■ - ■ where Q. (M) is the space of smooth functions on M, Q. (M) is the space of 1 -forms, and so forth. Forms which are the image of other forms under the exterior derivative are called exact and forms whose exterior derivative is are called closed (see closed and exact differential forms); the relationship ^ 2 — Qthen says that exact forms are closed. The converse, however, is not in general true; closed forms need not be exact. A simple but significant case is the 1-form of angle measure on the unit circle, written conventionally as d6. There is no actual function 6 defined on the whole circle for which this is true; the increment of 2jt in going once round the circle in the positive direction means that we can't take a single-valued 6. We can, however, change the topology by removing just one point. The idea of de Rham cohomology is to classify the different types of closed forms on a manifold. One performs this classification by saying that two closed forms a and |3 in f2 fc (JVf)are cohomologous if they differ by an exact form, that is, if Oi — (3 is exact. This classification induces an equivalence relation on the space of closed forms in D, k (M ) ■ One then defines the fa -th de Rham cohomology group H^ (M) to be the set of equivalence classes, that is, the set of closed forms in Q, k (M) modulo the exact forms. Note that, for any manifold M with n connected components H° dR (M) = R". This follows from the fact that any smooth function on M with zero derivative (i.e. locally constant) is constant on each of the connected components of M. De Rham cohomology computed One may often find the general de Rham cohomologies of a manifold using the above fact about the zero cohomology and a Mayer— Vietoris sequence. Another useful fact is that the de Rham cohomology is a homotopy invariant. While the computation is not given, the following are the computed de Rham cohomologies for some common topological objects: The n -sphere: For the n-sphere, and also when taken together with a product of open intervals, we have the following. Let n > 0, m > 0, and / an open real interval. Then i? d fe R (^xn-{ R i£k = > n > dlU ' [0 ifjfc#0,n. The n-torus: Similarly, allowing n > here, we obtain tf d fe R Cn^RC:). De Rham cohomology 219 Punctured Euclidean space: Punctured Euclidean space is simply Euclidean space with the origin removed. For n > 0, we have: i&(R"-{0}) (r iffc = 0,n-l ~ [0 if fc/ 0,n-l The Mobius strip, M: This more-or-less follows from the fact that the Mobius strip may be, loosely speaking, "contracted" to the 1 -sphere: De Rham's theorem Stokes' theorem is an expression of duality between de Rham cohomology and the homology of chains. It says that the pairing of differential forms and chains, via integration, gives a homomorphism from de Rham cohomology Hjr,(M) l ° singular cohomology groups H (M; R). De Rham's theorem, proved by Georges de Rham in 1931, states that for a smooth manifold M, this map is in fact an isomorphism. The wedge product endows the direct sum of these groups with a ring structure. A further result of the theorem is that the two cohomology rings are isomorphic (as graded rings), where the analogous product on singular cohomology is the cup product. Sheaf-theoretic de Rham isomorphism The de Rham cohomology is isomorphic to the Cech cohomology H (\J,F), where F is the sheaf of abelian groups determined by F(U) = R for all connected open sets U in M, and for open sets U and V such that U C V, the group morphism res : F(V) — > F(U) is given by the identity map on R, and where U is a good open cover of M (i.e. all the open sets in the open cover U are contractible to a point, and all finite intersections of sets in U are either empty or contractible to a point). Stated another way, if M is a compact C m manifold of dimension m, then for each k<m, there is an isomorphism i^ R (M,R) = H k (M,TL) where the left-hand side is the k-th de Rham cohomology group and the right-hand side is the Cech cohomology for the constant sheaf with fibre R. Proof k jfi + 1 Let Q, denote the sheaf of germs of fc-forms on M (with Q the sheaf of C functions on M). By the Poincare lemma, the following sequence of sheaves is exact (in the category of sheaves): o _> r _> q° i> fj 1 i» q 2 i» . . . i> rr -> o. This sequence now breaks up into short exact sequences o -> da 1 *- 1 ^ n k -i dn k -> o. Each of these induces a long exact sequence in cohomology. Since the sheaf of C m functions on a manifold i k admits partitions of unity, the sheaf-cohomology H (Q, ) vanishes for />0. So the long exact cohomology sequences themselves ultimately separate into a chain of isomorphisms. At one end of the chain is the Cech cohomology and at the other lies the de Rham cohomology. De Rham cohomology 220 Related ideas The de Rham cohomology has inspired many mathematical ideas, including Dolbeault cohomology, Hodge theory, and the Atiyah-Singer index theorem. However, even in more classical contexts, the theorem has inspired a number of developments. Firstly, the Hodge theorem proves that there is an isomorphism between the cohomology consisting of harmonic forms and the de Rham cohomology consisting of closed forms modulo exact forms. This relies on an appropriate definition of harmonic forms and of the Hodge theorem. For further details see Hodge theory. Harmonic forms If M is a compact Riemannian manifold, then each equivalence class in H%> (M) contains exactly one harmonic form. That is, every member to of a given equivalence class of closed forms can be written as u} = da + 7 where a is some form, and y is harmonic: Ay=0. Any harmonic function on a compact connected Riemannian manifold is a constant. Thus, this particular representative element can be understood to be an extremum (a minimum) of all cohomologously equivalent forms on the manifold. For example, on a 2-torus, one may envision a constant 1-form as one where all of the "hair" is combed neatly in the same direction (and all of the "hair" having the same length). In this case, there are two cohomologically distinct combings; all of the others are linear combinations. In particular, this implies that the 1st Betti number of a two-torus is two. More generally, on an n-dimensional torus 1 , one can consider the various combings of fc-forms on the torus. There are n choose k such combings that can be used to form the basis vectors for Hj-o (T n )', the k-th Betti number for the de Rham cohomology group for the n-torus is thus n choose k. More precisely, for a differential manifold M, one may equip it with some auxiliary Riemannian metric. Then the Laplacian A is defined by A = dS + 6d with d the exterior derivative and 5 the codifferential. The Laplacian is a homogeneous (in grading) linear differential operator acting upon the exterior algebra of differential forms: we can look at its action on each component of degree k separately. If M is compact and oriented, the dimension of the kernel of the Laplacian acting upon the space of k-forms is then equal (by Hodge theory) to that of the de Rham cohomology group in degree k: the Laplacian picks out a unique harmonic form in each cohomology class of closed forms. In particular, the space of all harmonic fc-forms on M is isomorphic to H (M;R). The dimension of each such space is finite, and is given by the k-th Betti number. Hodge decomposition Letting fi be the codifferential, one says that a form u) is co-closed if fi^ = and co-exact if u = Sa f° r some 2 form a . The Hodge decomposition states that any &-form can be split into three L components: u = da + 5j3 + 7 where 7is harmonic: A7 = 0. This follows by noting that exact and co-exact forms are orthogonal; the orthogonal complement then consists of forms that are both closed and co-closed: that is, of harmonic forms. Here, orthogonality is defined with respect to the L inner product on Q, k (M ) '■ (a : /3) = f aA*/3. Jm A precise definition and proof of the decomposition requires the problem to be formulated on Sobolev spaces. The idea here is that a Sobolev space provides the natural setting for both the idea of square-integrability and the idea of differentiation. This language helps overcome some of the limitations of requiring compact support. De Rham cohomology 221 References • Bott, Raoul; Tu, Loring W. (1982), Differential Forms in Algebraic Topology, Berlin, New York: Springer- Verlag, ISBN 978-0-387-90613-3 • Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, MR1288523, ISBN 978-0-471-05059-9 • Warner, Frank (1983), Foundations of Differentiable Manifolds and Lie Groups, Berlin, New York: Springer- Verlag, ISBN 978-0-387-90894-6 Algebraic geometry and analytic geometry In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties. Background Algebraic varieties are locally defined as the common zero sets of polynomials and since polynomials over the complex numbers are holomorphic functions, algebraic varieties over C can be interpreted as analytic spaces. Similarly, regular morphisms between varieties are interpreted as holomorphic mappings between analytic spaces. Somewhat surprisingly, it is often possible to go the other way, to interpret analytic objects in an algebraic way. For example, it is easy to prove that the analytic functions from the Riemann sphere to itself are either the rational functions or the identically infinity function (an extension of Liouville's theorem). For if such a function / is nonconstant, then since the set of z where f(z) is infinity is isolated and the Riemann sphere is compact, there are finitely many z with f(z) equal to infinity. Consider the Laurent expansion at all such z and subtract off the singular part: we are left with a function on the Riemann sphere with values in C, which by Liouville's theorem is constant. Thus /is a rational function. This fact shows there is no essential difference between the complex projective line as an algebraic variety, or as the Riemann sphere. Important results There is a long history of comparison results between algebraic geometry and analytic geometry, beginning in the nineteenth century and still continuing today. Some of the more important advances are listed here in chronological order. Riemann 's existence theorem Riemann surface theory shows that a compact Riemann surface has enough meromorphic functions on it, making it an algebraic curve. Under the name Riemann's existence theorem a deeper result on ramified coverings of a compact Riemann surface was known: such finite coverings as topological spaces are classified by permutation representations of the fundamental group of the complement of the ramification points. Since the Riemann surface property is local, such coverings are quite easily seen to be coverings in the complex-analytic sense. It is then possible to conclude that they come from covering maps of algebraic curves — that is, such coverings all come from finite extensions of the function field. Algebraic geometry and analytic geometry 222 The Lefschetz principle In the twentieth century, the Lefschetz principle, named for Solomon Lefschetz, was cited in algebraic geometry to justify the use of topological techniques for algebraic geometry over any algebraically closed field K of characteristic 0. by treating K as if it were the complex number field. It roughly asserts that true statements in algebraic geometry over C are true over any algebraically closed field K of characteristic zero. A precise principle and its proof are due to Alfred Tarski and are based in mathematical logic. This principle permits the carrying over of results obtained using analytic or topological methods for algebraic varieties over C to other algebraically closed ground fields of characteristic 0. Chow's theorem Chow's theorem, proved by W. L. Chow, is an example of the most immediately useful kind of comparison available. It states that an analytic subspace of complex projective space that is closed (in the ordinary topological sense) is an algebraic subvariety. This can be rephrased concisely as "any analytic subspace of complex projective space which is closed in the strong topology is closed in the Zariski topology." This allows quite a free use of complex-analytic methods within the classical parts of algebraic geometry. Serre's GAGA Foundations for the many relations between the two theories were put in place during the early part of the 1950s, as part of the business of laying the foundations of algebraic geometry to include, for example, techniques from Hodge theory. The major paper consolidating the theory was Geometrie Algebrique et Geometrie Analytique by Serre, now usually referred to as GAGA. It proves general results that relate classes of algebraic varieties, regular morphisms and sheaves with classes of analytic spaces, holomorphic mappings and sheaves. It reduces all of these to the comparison of categories of sheaves. Nowadays the phrase GAGA-style result is used for any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, to a well-defined subcategory of analytic geometry objects and holomorphic mappings. Formal statement of GAGA 1. Let (X, Ox)^ e a scheme of finite type over C. Then there is a topological space X m which as a set consists of the closed points of X with a continuous inclusion map X : X w — > X. The topology on X &a is called the "complex A. topology" (and is very different from the subspace topology). 2. Suppose cp: X — > Y is a morphism of schemes of locally finite type over C. Then there exists a continuous map ^an. ^an ^ yan such ^ ^an = ^ ^ 3. There is a sheaf (D^on X such that (X art , (D^ 1 ) is a ringed space and X : X — > X becomes a map of ringed spaces. The space (X an , (D^ 1 ) is called the "analytifiction" of (X , O x ) an d * s an analytic space. For every cp: X — > Y the map cp defined above is a mapping of analytic spaces. Furthermore, the map cp h- > cp maps open immersions into open immersions. If X= C[x ,...,x ] then X an = C" and Q^CU} for every polydisc £/ is a suitable quotient of the space of holomorphic functions, on U. ,„ 4. For every sheaf J^onX (called algebraic sheaf) there is a sheaf J? a7l on a (called analytic sheaf) and a map of sheaves of O x -modules \* x : T — >• {A x ) S: .F m . The sheaf J™ is defined as X^T ® x -^ O a ^ . The correspondence J? |_> J? an defines an exact functor from the category of sheaves over (X , Ox) t0 tne category of sheaves of (X an , O™) ■ The following two statements are the heart of Serre's GAGA theorem (as extended by Grothendieck, Neeman et al.) 5. Iff: X — > Y is an arbitrary morphism of schemes of finite type over C and ^"is coherent then the natural map (f*J-} an — > f m J^ an ' is injective. If/is proper then this map is an isomorphism. One also has isomorphisms of Algebraic geometry and analytic geometry 223 all higher direct image sheaves (J^ f^J 7 ) 0,71 = Jl' k f an j: an in this case. 6. Now assume that X is hausdorff and compact. If J-*, Q are two coherent algebraic sheaves on (X, O x )and if / : J- an — > Q an is a map of sheaves of C^ 1 modules then there exists a unique map of sheaves of O x modules ip : J- — > Q with/= cp . If 72. is a coherent analytic sheaf of (D°£ modules overX then there exists a coherent algebraic sheaf J? of (9x~ m °dules and an isomorphism J7 an ^ J^ . Moishezon manifolds A Moishezon manifold M is a compact connected complex manifold such that the field of meromorphic functions on M has transcendence degree equal to the complex dimension of M. Complex algebraic varieties have this property, but the converse is not (quite) true. The converse is true in the setting of algebraic spaces. In 1967, Boris Moishezon showed that a Moishezon manifold is a projective algebraic variety if and only if it admits a Kahler metric. Notes [1] For discussions see A. Seidenberg, Comments on Lefschetz's Principle, The American Mathematical Monthly, Vol. 65, No. 9 (Nov., 1958), pp. 685-690; 'Gerhard Frey and Hans-Georg Ruck, The strong Lefschetz principle in algebraic geometry, Manuscripta Mathematica, Volume 55, Numbers 3-4, September, 1986, pp. 385-401. [2] Hazewinkel, Michiel, ed. (2001), "Transfer principle" (http://eom.springer.de/T/tl 10050.htm), Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, References • Serre, Jean-Pierre (1956), "Geometrie algebrique et geometrie analytique" (http://www.numdam.org/ numdam-bin/item?id=AIF_1956 6 1_0), Universite de Grenoble. Annales de llnstitut Fourier 6: 1—42, MR0082175, ISSN 0373-0956 Riemannian manifold 224 Riemannian manifold In Riemannian geometry, a Riemannian manifold or Riemannian space (M,g) is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, in a manner which varies smoothly from point to point. The metric g is a positive definite symmetric tensor: a metric tensor. In other words, a Riemannian manifold is a differentiable manifold in which the tangent space at each point is a finite-dimensional Euclidean space. This allows one to define various geometric notions on a Riemannian manifold such as angles, lengths of curves, areas (or volumes), curvature, gradients of functions and divergence of vector fields. Riemannian manifolds should not be confused with Riemann surfaces, manifolds that locally appear like patches of the complex plane. The terms are named after German mathematician Bernhard Riemann. Overview The tangent bundle of a smooth manifold M assigns to each fixed point of M a vector space called the tangent space, and each tangent space can be equipped with an inner product. If such a collection of inner products on the tangent bundle of a manifold varies smoothly as one traverses the manifold, then concepts that were defined only pointwise at each tangent space can be extended to yield analogous notions over finite regions of the manifold. For example, a smooth curve a(t); [0, 1] — > M has tangent vector a'(t ) in the tangent space TM(f ) at any point t € (0, 1), and each such vector has length la'(f )ll, where Ml denotes the norm induced by the inner product on TM(f ). The integral of these lengths gives the length of the curve a: L(a) = J \\a'(t)\\dt. Jo Smoothness of a(f) for t in [0, 1] guarantees that the integral L(a) exists and the length of this curve is defined. In many instances, in order to pass from a linear-algebraic concept to a differential-geometric one, the smoothness requirement is very important. Every smooth submanifold of R" has an induced Riemannian metric g: the inner product on each tangent space is the restriction of the inner product on R . In fact, as follows from the Nash embedding theorem, all Riemannian manifolds can be realized this way. In particular one could define Riemannian manifold as a metric space which is isometric to a smooth submanifold of R" with the induced intrinsic metric, where isometry here is meant in the sense of preserving the length of curves. This definition might theoretically not be flexible enough, but it is quite useful to build the first geometric intuitions in Riemannian geometry. Riemannian manifolds as metric spaces Usually a Riemannian manifold is defined as a smooth manifold with a smooth section of the positive-definite quadratic forms on the tangent bundle. Then one has to work to show that it can be turned to a metric space: If y: [a, b] — > M is a continuously differentiable curve in the Riemannian manifold M, then we define its length L(y) in analogy with the example above by m= I iiYwiid*. J a With this definition of length, every connected Riemannian manifold M becomes a metric space (and even a length metric space) in a natural fashion: the distance d(x, y) between the points x and y of M is defined as d(x,y) = inf{ L(y) : y is a continuously differentiable curve joining x and y}. Riemannian manifold 225 Even though Riemannian manifolds are usually "curved," there is still a notion of "straight line" on them: the geodesies. These are curves which locally join their points along shortest paths. Assuming the manifold is compact, any two points x and y can be connected with a geodesic whose length is d(x,y). Without compactness, this need not be true. For example, in the punctured points (-1,0) and (1, 0) is 2, but there is no geodesic realizing this distance. 2 Without compactness, this need not be true. For example, in the punctured plane R \ {0}, the distance between the Properties In Riemannian manifolds, the notions of geodesic completeness, topological completeness and metric completeness are the same: that each implies the other is the content of the Hopf-Rinow theorem. Riemannian metrics Let M be a differentiable manifold of dimension n. A Riemannian metric on M is a family of (positive definite) inner products g p : T P M x T P M — ► R, p e M such that, for all differentiable vector fields X,Y on M, p^g p (X(p),Y(p)) defines a smooth function M — > R. More formally, a Riemannian metric Q is a section of the vector bundle S 2 T*M- 12 n In a system of local coordinates on the manifold M given by n real-valued functions x ,x , . . . , x , the vector fields f o d l [ dx 1 ' ' dx n j give a basis of tangent vectors at each point of M. Relative to this coordinate system, the components of the metric tensor are, at each point p, 9M := 9 \{ir) ; { a dxi ' p . 1 n Equivalently , the metric tensor can be written in terms of the dual basis { dx ,..., dx } of the cotangent bundle as 9 = / gij&x 1 ® dx 3 . Endowed with this metric, the differentiable manifold (M,g) is a Riemannian manifold. Examples • With -identified with e^ = (0, . . . , 1, . . . , 0), the standard metric over an open subset JJ (2 R^is dx defined by 9 ™:T p UxT p U^R, (E^'E^) ^E a ^ \ i j / i Then g is a Riemannian metric, and Equipped with this metric, R is called Euclidean space of dimension n and g.. is called the Euclidean metric. Riemannian manifold 226 • Let (M,g) be a Riemannian manifold and ]\[ (^ JVf be a submanifold of M. Then the restriction of g to vectors tangent along N defines a Riemannian metric over N. • More generally, let/:M n — ^jV" be an immersion. Then, if N has a Riemannian metric, /induces a Riemannian metric on M via pullback: 5 f : T p M x T p M — ► R, This is then a metric; the positive definiteness follows of the injectivity of the differential of an immersion. Let (M,g ) be a Riemannian manifold, h:M — »iV be a differentiable map and q€N be a regular value of h (the differential d/j(/?) is surjective for all /?€/*" (<jr)). Then K (q)CM is a submanifold of M of dimension n. Thus /T (^) carries the Riemannian metric induced by inclusion. In particular, consider the following map : n h-.W 1 — ►R, (x 1 ,...,x n )\ — >J^(<) 2 -1. i=l Then, is a regular value of h and 71 /T^O) = {z£ R n | J^) 2 = 1} = S 71 ' 1 is the unit sphere g 71-1 (^ ][£"•. The metric induced from R"on g n — ^is called the canonical metric of £71-1. Let Afi^nd M.2^ two Riemannian manifolds and consider the cartesian product M\ X A^with the product structure. Furthermore, let tti : M± X A/2 — > Afiand 7T2 : Mi X Af 2 — *■ Af^be the natural projections. For (.Pi <?) £ Mi X M2, a Riemannian metric on Afi X Af-jcan be introduced as follows : MixM 2 C) : W M i x Ms ) x W M i x Ma ) — * R ' («,v) 1 — > g^iT^Tv^u)^^^)) + g^ 2 (T^ q} 7v 2 (u),T {Ptq) 7r 2 (v)). Ci^M^T^M^ + C The identification T M {M 1 x M 2 ) = T^ © T g M 2 allows us to conclude that this defines a metric on the product space. The torus S 1 X X S^ = T n possesses for example a Riemannian structure obtained by choosing the induced Riemannian metric from J| 2 on the circle g 1 q ]fj 2 and then taking the product metric. The torus r J m endowed with this metric is called the flat torus. Let go , C/i be two metrics on j\^f . Then, g := Xg + (1 - \)g u AG [0,1], is also a metric on M. Riemannian manifold 227 The pullback metric If f.M-^N is a differentiable map and (N,g ) a Riemannian manifold, then the pullback of g along /is a quadratic form on the tangent space of M. The pullback is the quadratic form f*g on TM defined for v, w G T M by (rg N )(v,w)=g N (df(v),df(w)). where df(v) is the pushforward of v by/. The quadratic form f*g N is in general only a semi definite form because df can have a kernel. If / is a diffeomorphism, or more generally an immersion, then it defines a Riemannian metric on M, the pullback metric. In particular, every embedded smooth submanifold inherits a metric from being embedded in a Riemannian manifold, and every covering space inherits a metric from covering a Riemannian manifold. Existence of a metric Every paracompact differentiable manifold admits a Riemannian metric. To prove this result, let M be a manifold and {(U , (p(U ))la€/} a locally finite atlas of open subsets U of M and diffeomorphisms onto open subsets of R <p:U a ^ (j)(U a ) C R n . Let x be a differentiable partition of unity subordinate to the given atlas. Then define the metric g on M by 9 ■= ^2 t p ■ ^' with ^ := ^ can - where g is the Euclidean metric. This is readily seen to be a metric on M. Isometries Let (M n^)and (JV Q N )^° e two Riemannian manifolds, and f : M — > iVbe a diffeomorphism. Then, / is called an isometry, if 9 M = f*9 N , or pointwise flf («, «) = <4,)(T P /K>, T p /(i;)) Vp G M, Vzx, « € T p M. Moreover, a differentiable mapping / : M — > Nis called a local isometry at pG M if there is a neighbourhood JJ (2 M. , U 9 p, such that f : U — > f(U)is a diffeomorphism satisfying the previous relation. Riemannian manifolds as metric spaces A connected Riemannian manifold carries the structure of a metric space whose distance function is the arclength of a minimizing geodesic. Specifically, let (M,g) be a connected Riemannian manifold. Let c : [a, b] — > Af be a parametrized curve in M, which is differentiable with velocity vector c'. The length of c is defined as L b a (c):= J Vff(C(*),C(t))dt= J ||c'(i)||di. By change of variables, the arclength is independent of the chosen parametrization. In particular, a curve [a, b] — > .Mean be parametrized by its arc length. A curve is parametrized by arclength if and only if ||c'(i)|| = lforallf e [a,b]. The distance function d : MxM — > [0,°°) is defined by d(p,q) = infL(7) where the infimum extends over all differentiable curves y beginning at p€M and ending at q€M. Riemannian manifold 228 This function d satisfies the properties of a distance function for a metric space. The only property which is not completely straightforward is to show that d(p,q)=0 implies that p=q. For this property, one can use a normal coordinate system, which also allows one to show that the topology induced by d is the same as the original topology onM. Diameter The diameter of a Riemannian manifold M is defined by diam(M) := sup d(p, q) G R> U {+00}. p,q£M The diameter is invariant under global isometries. Furthermore, the Heine-Borel property holds for (finite-dimensional) Riemannian manifolds: M is compact if and only if it is complete and has finite diameter. Geodesic completeness A Riemannian manifold M is geodesically complete if for all p £j M , the exponential map ex P p is defined for all V (E T p M , i.e. if any geodesic "f(i) starting from p is defined for all values of the parameter t £ R • The Hopf-Rinow theorem asserts that M is geodesically complete if and only if it is complete as a metric space. If M is complete, then M is non-extendable in the sense that it is not isometric to an open proper submanifold of any other Riemannian manifold. The converse is not true, however: there exist non-extendable manifolds which are not complete. See also Riemannian geometry Finsler manifold sub-Riemannian manifold pseudo-Riemannian manifold Metric tensor Hermitian manifold Space (mathematics) References • Jost, Jiirgen (2008), Riemannian Geometry and Geometric Analysis (5th ed.), Berlin, New York: Springer-Verlag, ISBN 978-3540773405 • do Carmo, Manfredo (1992), Riemannian geometry, Basel, Boston, Berlin: Birkhauser, ISBN 978-0-8176-3490-2 [1] External links • L.A. Sidorov (2001), "Riemannian metric" , in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104 Riemannian manifold 229 References [1] http://www.amazon.fr/Riemannian-Geometry-Manfredo-P-Carmo/dp/0817634908/ref=sr_l_l?ie=UTF8&s=english-books& qid=1201537059&sr=8-l [2] http://eom. springer. de/R/r082 1 80. htm List of complex analysis topics This is a list of complex analysis topics, by Wikipedia page. Local theory Holomorphic function Antiholomorphic function Cauchy-Riemann equations Conformal mapping Power series Radius of convergence Laurent series Meromorphic function Entire function Pole (complex analysis) Zero (complex analysis) Residue (complex analysis) Isolated singularity Removable singularity Essential singularity Branch point Principal branch Weierstrass-Casorati theorem Landau's constants Holomorphic functions are analytic Schwarzian derivative Analytic capacity Disk algebra Growth • Bieberbach conjecture • Borel-Caratheodory theorem • Hadamard three-circle theorem • Hardy space • Hardy's theorem • Progressive function • Corona theorem • Maximum modulus principle • Nevanlinna theory • Picard's theorem List of complex analysis topics 230 • Paley-Wiener theorem • Value distribution theory of holomorphic functions Contour integrals Line integral Cauchy integral theorem Cauchy's integral formula Residue theorem Liouville's theorem (complex analysis) Examples of contour integration Fundamental theorem of algebra Simply connected Winding number • Principle of the argument • Rouche's theorem • Bromwich integral • Morera's theorem • Mellin transform • Kramers— Kronig relation Special functions • Exponential function • Beta function • Gamma function • Riemann zeta function • Riemann hypothesis • Generalized Riemann hypothesis • Elliptic function • Half-period ratio • Jacobi's elliptic functions • Weierstrass's elliptic functions • Theta function • Elliptic modular function • J-function • Modular function • Modular form List of complex analysis topics 23 1 Riemann surfaces • Analytic continuation • Riemann sphere • Riemann surface • Riemann mapping theorem • Caratheodory's theorem (conformal mapping) • Riemann-Roch theorem Other Antiderivative (complex analysis) Bocher's theorem Cayley transform Complex differential equation Harmonic conjugate Method of steepest descent Montel's theorem Periodic points of complex quadratic mappings Pick matrix Runge approximation theorem Schwarz lemma Weierstrass factorization theorem Mittag-Leffler's theorem Several complex variables Analytization trick Biholomorphy Cartan's theorems A and B Cousin problems Edge-of-the-wedge theorem Several complex variables History (needs work) • Augustin Louis Cauchy • Jacques Hadamard • Kiyoshi Oka • Bernhard Riemann • Karl Weierstrass 232 Algebraic Topology and Groupoids Algebraic topology Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. The method of algebraic invariants An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones (the modern standard tool for such construction is the CW-complex). The basic method now applied in algebraic topology is to investigate spaces via algebraic invariants by mapping them, for example, to groups which have a great deal of manageable structure in a way that respects the relation of homeomorphism (or more general homotopy) of spaces. This allows one to recast statements about topological spaces into statements about groups, which are often easier to prove. Two major ways in which this can be done are through fundamental groups, or more generally homotopy theory, and through homology and cohomology groups. The fundamental groups give us basic information about the structure of a topological space, but they are often nonabelian and can be difficult to work with. The fundamental group of a (finite) simplicial complex does have a finite presentation. Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated. Finitely generated abelian groups are completely classified and are particularly easy to work with. Setting in category theory In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural transformation originated here. Fundamental groups and homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms also correspond — a continuous mapping of spaces induces a group homomorphism on the associated groups, and these homomorphisms can be used to show non-existence (or, much more deeply, existence) of mappings. Results on homology Several useful results follow immediately from working with finitely generated abelian groups. The free rank of the n-th homology group of a simplicial complex is equal to the n-th Betti number, so one can use the homology groups of a simplicial complex to calculate its Euler-Poincare characteristic. As another example, the top-dimensional integral homology group of a closed manifold detects orientability: this group is isomorphic to either the integers or 0, according as the manifold is orientable or not. Thus, a great deal of topological information is encoded in the homology of a given topological space. Beyond simplicial homology, which is defined only for simplicial complexes, one can use the differential structure of smooth manifolds via de Rham cohomology, or Cech or sheaf cohomology to investigate the solvability of Algebraic topology 233 differential equations defined on the manifold in question. De Rham showed that all of these approaches were interrelated and that, for a closed, oriented manifold, the Betti numbers derived through simplicial homology were the same Betti numbers as those derived through de Rham cohomology. This was extended in the 1950s, when Eilenberg and Steenrod generalized this approach. They defined homology and cohomology as functors equipped with natural transformations subject to certain axioms (e.g., a weak equivalence of spaces passes to an isomorphism of homology groups), verified that all existing (co)homology theories satisfied these axioms, and then proved that such an axiomatization uniquely characterized the theory. A new approach uses a functor from filtered spaces to crossed complexes defined directly and homotopically using relative homotopy groups; a higher homotopy van Kampen theorem proved for this functor enables basic results in algebraic topology, especially on the border between homology and homotopy, to be obtained without using singular homology or simplicial approximation. This approach is also called nonabelian algebraic topology, and generalises to higher dimensions ideas coming from the fundamental group. Applications of algebraic topology Classic applications of algebraic topology include: • The Brouwer fixed point theorem: every continuous map from the unit n-disk to itself has a fixed point. • The n-sphere admits a nowhere-vanishing continuous unit vector field if and only if n is odd. (For « = 2, this is sometimes called the "hairy ball theorem".) • The Borsuk— Ulam theorem: any continuous map from the n-sphere to Euclidean n-space identifies at least one pair of antipodal points. • Any subgroup of a free group is free. This result is quite interesting, because the statement is purely algebraic yet the simplest proof is topological. Namely, any free group G may be realized as the fundamental group of a graph X. The main theorem on covering spaces tells us that every subgroup H of G is the fundamental group of some covering space 7 of X; but every such Y is again a graph. Therefore its fundamental group H is free. On the other hand this type of application is also handled by the use of covering morphisms of groupoids, and that technique has yielded subgroup theorems not yet proved by methods of algebraic topology (see the book by Higgins listed under groupoids). • Topological combinatorics Notable algebraic topologists Frank Adams Karol Borsuk Luitzen Egbertus Jan Brouwer William Browder Ronald Brown (mathematician) Nicolas Bourbaki Henri Cartan Hermann Kiinneth Samuel Eilenberg Hans Freudenthal Peter Freyd Alexander Grothendieck Friedrich Hirzebruch Heinz Hopf Michael J. Hopkins Algebraic topology 234 Witold Hurewicz Egbert van Kampen Saunders Mac Lane Jean Leray Mark Mahowald J. Peter May John Coleman Moore Sergei Novikov Lev Pontryagin Mikhail Postnikov Daniel Quillen Jean-Pierre Serre Stephen Smale Norman Steenrod Dennis Sullivan Rene Thom Leopold Vietoris Hassler Whitney J. H. C. Whitehead Brandon Blaha Important theorems in algebraic topology Borsuk-Ulam theorem Brouwer fixed point theorem Cellular approximation theorem Eilenberg— Zilber theorem Freudenthal suspension theorem Hurewicz theorem Kunneth theorem Poincare duality theorem Universal coefficient theorem Van Kampen's theorem Generalized van Kampen's theorems T21 Higher homotopy, generalized van Kampen's theorem Whitehead's theorem Notes [i] http://pianetphysics.org/encyciopedia/GeneraiizedvanKampenTheoremsHDGVKT.htmi#BHKP [2] R. Brown, K. A. Hardie, K. H. Kamps and T. Porter, A homotopy double groupoid of a Hausdorff space, Theory and Applications of Categories. 10(2002) 71-93. http://www.emis.de/journals/TAC/volumes/14/9/14-09.pdf References • Bredon, Glen E. (1993), Topology and Geometry (http://books.google.com/?id=G74V6UzL_PUC& printsec=frontcover&dq=bredon+topology+and+geometry), Graduate Texts in Mathematics 139, Springer, ISBN 0-387-97926-3, retrieved 2008-04-01. Algebraic topology 235 • Hatcher, Allen (2002), Algebraic Topology (http://www.math.cornell.edu/~hatcher/AT/ATpage.html), Cambridge: Cambridge University Press, ISBN 0-521-79540-0. A modern, geometrically flavored introduction to algebraic topology. • Maunder, C. R. F. (1970), Algebraic Topology, London: Van Nostrand Reinhold, ISBN 0-486-69131-4. • R. Brown and A. Razak, "A van Kampen theorem for unions of non-connected spaces, Archiv. Math. 42 (1984) 85-88. • P. J. Higgins, Categories and groupoids (1971) Van Nostrand-Reinhold. (http://138.73.27.39/tac/reprints/ articles/7/tr7abs.html) • Ronald Brown, Higher dimensional group theory (http://www.bangor.ac.Uk/r.brown/hdaweb2.html) (2007) (Gives a broad view of higher dimensional van Kampen theorems involving multiple groupoids). • E. R. van Kampen. On the connection between the fundamental groups of some related spaces. American Journal of Mathematics, vol. 55 (1933), pp. 261-267. • R. Brown, P.J. Higgins, and R. Sivera. Non-Abelian Algebraic Topology: filtered spaces, crossed complexes, cubical higher homotopy groupoids (http://www.bangor.ac.uk/~mas010/nonab-a-t.html); European Mathematical Society Tracts in Mathematics Vol. 15, 2010, downloadable PDF: (http://www.bangor.ac.uk/ ~mas010/pdffiles/rbrsbookb-e090809.pdf) • Van Kampen's theorem (http://planetmath.org/?op=getobj&from=objects&id=3947) on PlanetMath • Van Kampen's theorem result (http://planetmath.org/?op=getobj&from=objects&id=5576) on PlanetMath • Ronald Brown R, K. Hardie, H. Kamps, T. Porter T.: The homotopy double groupoid of a Hausdorff space., Theory Appl. Categories, 10:71—93 (2002). • Dylan G. L. Allegretti, Simplicial Sets and van Kampen's Theorem (http://www.math.uchicago.edu/~may/ VIGRE/VIGREREU2008.html) (Discusses generalized versions of van Kampen's theorem applied to topological spaces and simplicial sets). Further reading • Allen Hatcher, Algebraic topology. (http://www.math.cornell.edu/~hatcher/AT/ATpage.html) (2002) Cambridge University Press, Cambridge, xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0. • May, J. P. (1999), A Concise Course in Algebraic Topology (http://www.math.uchicago.edu/~may/ CONCISE/ConciseRevised.pdf), U. Chicago Press, Chicago, retrieved 2008-09-27. (Section 2.7 provides a category-theoretic presentation of the theorem as a colimit in the category of groupoids). • Higher dimensional algebra • Ronald Brown, Philip J. Higgins and Rafael Sivera. 2009. Higher dimensional, higher homotopy, generalized van Kampen Theorem., in Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical higher homotopy groupoids. 512 pp, (Preprint), (http://www.bangor.ac.uk/~mas010/pdffiles/rbrsbookb-e090809. pdf) • Ronald Brown, Topology and groupoids (http://www.bangor.ac.Uk/r.brown/topgpds.html) (2006) Booksurge LLC ISBN 1-4196-2722-8. Groupoid 236 Groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: • Group with a partial function replacing the binary operation; • Category in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation, called inverse by analogy with group theory. Special cases include: • Setoids, that is: sets which come with an equivalence relation; • G-sets, sets equipped with an action of a group G. Groupoids are often used to reason about geometrical objects such as manifolds. Heinrich Brandt introduced groupoids implicitly via Brandt semigroups in 1926. Definitions Algebraic A groupoid is a set G with a unary operation — 1 ; Q — ^ Q and a partial function * ; G X G — > G.* is not a binary operation because it is not necessarily defined for all possible pairs of G-elements. The precise conditions under which * is defined are not articulated here and vary by situation. • and " have the following axiomatic properties. Let a, b, and c be elements of G. Then: • Associativity: If a * b and b * c are defined, then (a* b) * c and a * (b * c) are defined and equal. Conversely, if either of these last two expressions is defined, then so is the other (and again they are equal). • Inverse: a' * a and a * a' are always defined. • Identity: If a * b is defined, then a * b * b~ = a, and a' * a* b = b. (The previous two axioms already show that these expressions are defined and unambiguous.) In short: • (a * b) * c = a * (b * c); • (a * b) * b = a; • a * (a * b) = b. From these axioms, two easy and convenient theorems follow: • (a ) = a; • If a * b is defined, then (a * b)~ = b~ * a' . Category theoretic A groupoid is a small category in which every morphism is an isomorphism, and hence invertible. More precisely, a groupoid G is: • A set G of objects; • For each pair of objects x and y in G , there exists a (possibly empty) set G(x,y) of morphisms (or arrows) from x to y. We write/: x — > y to indicate that/is an element of G(x,y). The objects and morphisms have the properties: • For every object x, there exists the element id^ of G(x,x); • For each triple of objects x, y, and z, there exists the function com P x ,y,z ■ G(x,y) x G{y,z) — > G(x,z). We write gf Tor comp xy z (f, g) , where/ £ G(x,y), and g £ G(y,z); Groupoid 237 • There exists the function inv^ y : G(x,y) — > G(y,x). Moreover, iff : x — » y, g : y — > z, and h : z — > w, then: • /ids = /andid y / = /; • (hg)f=h(gf); ' /inv(/) = id y and inv(/)/ = id x . If /is an element of G(x,y) then x is called the source off written s(f), and y the target of/ (written t(f)). Comparing the definitions The algebraic and category-theoretic definitions are equivalent, as follows. Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G(x,y) (i.e. the sets of morphisms from x to y). Then compand invbecome partially defined operations on G, and inv w iH m f act be defined everywhere; so we define * to be compand — lto be uiv Thus we have a groupoid in the algebraic sense. Explicit reference to G (and hence to jd) can be dropped. Conversely, given a groupoid G in the algebraic sense, with typical element/ let G be the set of all elements of the form/ 1 /" . In other words, the objects are identified with the identity morphisms, so that id x is just x. Let G(x,y) be the set of all elements /such that yfx is defined. Then " and * break up into several functions on the various G(x,y), which may be called inv an d comp, respectively. Sets in the definitions above may be replaced with classes, as is generally the case in category theory. Vertex groups Given a groupoid G, the vertex groups or isotropy group in G are the subsets of the form G(x,x), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group. Category of groupoids The category whose objects are groupoids and whose morphisms are groupoid homomorphisms is called the groupoid category, or the category of groupoids. Examples Linear algebra Given a field K, the corresponding general linear groupoid GLJ.K) consists of all invertible matrices whose entries range over K. Matrix multiplication interprets composition. If G = GL^K), then the set of natural numbers is a proper subset of G , since for each natural number n, there is a corresponding identity matrix of dimension n. G(m,n) is empty unless m=n, in which case it is the set of all nxn invertible matrices. Topology Given a topological space X, let G be the set X. The morphisms from the point p to the point q are equivalence classes of continuous paths from/? to q, with two paths being equivalent if they are homotopic. Two such morphisms are composed by first following the first path, then the second; the homotopy equivalence guarantees that this composition is associative. This groupoid is called the fundamental groupoid of X, denoted TTi(X), The usual fundamental group 7^ (X, a;) is then the vertex group for the point x. An important extension of this idea is to consider the fundamental groupoid 7Ti(X,A) where A is a set of "base points" and a subset of X Here, one considers only paths whose endpoints belong to A. 7Ti(XA) is a sub-groupoid of 7Ti(X). The set A may be chosen according to the geometry of the situation at hand. Groupoid 238 Equivalence relation If X is a set with an equivalence relation denoted by infix relation can be formed as follows: then a groupoid "representing" this equivalence • The objects of the groupoid are the elements of X; • For any two elements x and y in X, there is a single morphism from x to y if and only if x~y. Group action If the group G acts on the set X, then we can form the action groupoid representing this group action as follows: • The objects are the elements of X; • For any two elements x and y in X, there is a morphism from x to y corresponding to every element g of G such that gx = y; • Composition of morphisms interprets the binary operation of G. More explicitly, the action groupoid is the set Q x yf with source and target maps s(g,x) = x and t(g,x) = gx. It is often denoted G K X( ov X X G )■ Multiplication (or composition) in the groupoid is then (h,y)(g,x) — (hg, x) which is defined provided y-gx. For x in X, the vertex group consists of those (g,x) with gx = x, which is just the isotropy subgroup at x for the given action (which is why vertex groups are also called isotropy groups). Another way to describe G-sets is the functor category [Gr, Set] , where Gri s the groupoid (category) with one element and isomorphic to the group G. Indeed, every functor F of this category defines a set X=F (Gr) and for every g in G (i.e. for every morphism in Gr) induces a bijection F : X—*X, The categorical structure of the functor F assures us that F defines a G-action on the set X. The (unique) representable functor F : Gr - * Seti s the Cay ley representation of G. In fact, this functor is isomorphic to HomfGr, — )and so sends ob(Gr)to the set Hom(Gr, Gr)which is by definition the "set" G and the morphism g of Gr(i- e - the element g of G) to the permutation F of the set G. We deduce from the Yoneda embedding that the group G is isomorphic to the group {F I g€G}, a subgroup of the group of permutations of G Fifteen puzzle The symmetries of the fifteen puzzle form a groupoid (not a group, as not all moves can be composed) groupoid acts on configurations. [2] [3] This Relation to groups Group-like structures Totality Associativity Identity Inverses Group Yes Yes Yes Yes Monoid Yes Yes Yes No Semigroup Yes Yes No No Loop Yes No Yes Yes Quasigroup Yes No No Yes Magma Yes No No No Groupoid No Yes Yes Yes Category No Yes Yes No Groupoid 239 If a groupoid has only one object, then the set of its morphisms forms a group. Using the algebraic definition, such a groupoid is literally just a group. Many concepts of group theory generalize to groupoids, with the notion of functor replacing that of group homomorphism. If x is an object of the groupoid G, then the set of all morphisms from x to x forms a group G(x). If there is a morphism /from x to y, then the groups G(x) and G(y) are isomorphic, with an isomorphism given by the mapping g Every connected groupoid (that is, one in which any two objects are connected by at least one morphism) is isomorphic to a groupoid of the following form. Pick a group G and a set (or class) X. Let the objects of the groupoid be the elements of X. For elements x and y of X, let the set of morphisms from x to y be G. Composition of morphisms is the group operation of G. If the groupoid is not connected, then it is isomorphic to a disjoint union of groupoids of the above type (possibly with different groups G for each connected component). Thus any groupoid may be given (up to isomorphism) by a set of ordered pairs (X,G). Note that the isomorphism described above is not unique, and there is no natural choice. Choosing such an isomorphism for a connected groupoid essentially amounts to picking one object x , a group isomorphism h from G(x ) to G, and for each x other than x_, a morphism in G from x to x. In category-theoretic terms, each connected component of a groupoid is equivalent (but not isomorphic) to a groupoid with a single object, that is, a single group. Thus any groupoid is equivalent to a multiset of unrelated groups. In other words, for equivalence instead of isomorphism, one need not specify the sets X, only the groups G. Consider the examples in the previous section. The general linear groupoid is both equivalent and isomorphic to the disjoint union of the various general linear groups GL (F). On the other hand: • The fundamental groupoid of X is equivalent to the collection of the fundamental groups of each path-connected component of X, but an isomorphism requires specifying the set of points in each component; • The set X with the equivalence relation ~ is equivalent (as a groupoid) to one copy of the trivial group for each equivalence class, but an isomorphism requires specifying what each equivalence class is: • The set X equipped with an action of the group G is equivalent (as a groupoid) to one copy of G for each orbit of the action, but an isomorphism requires specifying what set each orbit is. The collapse of a groupoid into a mere collection of groups loses some information, even from a category-theoretic point of view, because it is not natural. Thus when groupoids arise in terms of other structures, as in the above examples, it can be helpful to maintain the full groupoid. Otherwise, one must choose a way to view each G(x) in terms of a single group, and this choice can be arbitrary. In our example from topology, you would have to make a coherent choice of paths (or equivalence classes of paths) from each point p to each point q in the same path-connected component. As a more illuminating example, the classification of groupoids with one endomorphism does not reduce to purely group theoretic considerations. This is analogous to the fact that the classification of vector spaces with one endomorphism is nontrivial. Morphisms of groupoids come in more kinds than those of groups: we have, for example, fibrations, covering morphisms, universal morphisms, and quotient morphisms. Thus a subgroup H of a group G yields an action of G on the set of cosets of H in G and hence a covering morphism p from, say, K to G, where K is a groupoid with vertex groups isomorphic to H. In this way, presentations of the group G can be "lifted" to presentations of the groupoid K, and this is a useful way of obtaining information about presentations of the subgroup H. For further information, see the books by Higgins and by Brown in the References. Another useful fact is that the category of groupoids, unlike that of groups, is cartesian closed. Groupoid 240 Lie groupoids and Lie algebroids When studying geometrical objects, the arising groupoids often carry some differentiable structure, turning them into Lie groupoids. These can be studied in terms of Lie algebroids, in analogy to the relation between Lie groups and Lie algebras. Notes [1] Brandt semigroup (http://eom. springer. de/b/bO 17600. htm) in Springer Encyclopaedia of Mathematics - ISBN 1-4020-0609-8 [2] The 15-puzzle groupoid (1) (http://www.neverendingbooks.org/index.php/the-15-puzzle-groupoid-l.html), Never Ending Books [3] The 15-puzzle groupoid (2) (http://www.neverendingbooks.org/index.php/the-15-puzzle-groupoid-2.html), Never Ending Books References • Brown, Ronald, 1987, " From groups to groupoids: a brief survey, (http://www.bangor.ac.Uk/r.brown/ groupoidsurvey.pdf)" Bull. London Math. Soc. 19: 113-34. Reviews the history of groupoids up to 1987, starting with the work of Brandt on quadratic forms. The downloadable version updates the many references. • , 2006. Topology and groupoids. (http://www.bangor.ac.Uk/r.brown/topgpds.html) Booksurge. Revised and extended edition of a book previously published in 1968 and 1988. Groupoids are introduced in the context of their topological application. • , Higher dimensional group theory (http://www.bangor.ac.Uk/r.brown/hdaweb2.htm) Explains how the groupoid concept has to led to higher dimensional homotopy groupoids, having applications in homotopy theory and in group cohomology. Many references. • F. Borceux, G Janelidze, 2001, Galois theories, (http://www.cup.cam.ac.uk/catalogue/catalogue. asp?isbn=978052 1803090) Cambridge Univ. Press. Shows how generalisations of Galois theory lead to Galois groupoids. • Cannas da Silva, A., and A. Weinstein, Geometric Models for Noncommutative Algebras, (http://www.math.ist. utl.pt/~acannas/Books/models_final.pdf) Especially Part VI. • Golubitsky, M., Ian Stewart, 2006, " Nonlinear dynamics of networks: the groupoid formalism (http://www. ams.org/bull/2006-43-03/S0273-0979-06-01108-6/S0273-0979-06-01108-6.pdf)", Bull. Amer. Math. Soc. 43: 305-64 • Higgins, P. J., "The fundamental groupoid of a graph of groups", J. London Math. Soc. (2) 13 (1976) 145—149. • Higgins, P. J. and Taylor, J., "The fundamental groupoid and the homotopy crossed complex of an orbit space", in Category theory (Gummersbach, 1981), Lecture Notes in Math., Volume 962. Springer, Berlin (1982), 115—122. • Higgins, P. J., 1971. Categories and groupoids. Van Nostrand Notes in Mathematics. Republished in Reprints in Theory and Applications of Categories, No. 7 (2005) pp. 1-195; freely downloadable (http://www.tac.mta.ca/ tac/reprints/articles/7/tr7abs.html). Substantial introduction to category theory with special emphasis on groupoids. Presents applications of groupoids in group theory, for example to a generalisation of Grushko's theorem, and in topology, e.g. fundamental groupoid. • Mackenzie, K. C. H, 2005. General theory of Lie groupoids and Lie algebroids. (http://www.shef.ac.uk/ -pmlkchm/gt.html) Cambridge Univ. Press. • Weinstein, Alan, " Groupoids: unifying internal and external symmetry — A tour though some examples, (http:// www.ams.org/notices/199607/weinstein.pdf)" Also available in Postscript, (http://math.berkeley.edu/ -alanw/Groupoids.ps), Notices of AMS, July 1996, pp. 744-752. Galois group 241 Galois group In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions (and polynomials which give rise to them) via Galois groups is called Galois theory, so named in honor of Evariste Galois who first discovered them. For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory. Definition Suppose that E is an extension of the field F (written as EIF and read E over F), Consider the set of all automorphisms of EIF (that is, isomorphisms a from E to itself such that a(x) = x for every x in F). This set of automorphisms with the operation of function composition forms a group, sometimes denoted by A\xi{EIF). If EIF is a Galois extension, then A\xi{EIF) is called the Galois group of (the extension) E over F, and is usually denoted by Gal(£/F). [1] Examples In the following examples F is a field, and C, R, Q are the fields of complex, real, and rational numbers, respectively. The notation F{a) indicates the field extension obtained by adjoining an element a to the field F. • Gal(FIF) is the trivial group that has a single element, namely the identity automorphism. • Gal(C/R) has two elements, the identity automorphism and the complex conjugation automorphism. • Aut(R/Q) is trivial. Indeed it can be shown that any Q-automorphism must preserve the ordering of the real numbers and hence must be the identity. • Aut(C/Q) is an infinite group. • Gal(Q(V2)/Q) has two elements, the identity automorphism and the automorphism which exchanges V2 and -a/2. • Consider the field K = Q( 3 V2). The group Aut(K/Q) contains only the identity automorphism. This is because K is not a normal extension, since the other two cube roots of 2 (both complex) are missing from the extension — in other words K is not a splitting field. • Consider now L = Q( 3 V2, co), where co is a primitive third root of unity. The group Gal(L/Q) is isomorphic to S , 3 the dihedral group of order 6, and L is in fact the splitting field of x - 2 over Q. • If q is a prime power, and if F = G¥{q) and E = G¥{q ) denote the Galois fields of order q and q respectively, then Gal(E/F) is cyclic of order n. Properties The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory: the closed (with respect to the Krull topology below) subgroups of the Galois group correspond to the intermediate fields of the field extension. If EIF is a Galois extension, then Gal(£7F) can be given a topology, called the Krull topology, that makes it into a profinite group. Galois group 242 Notes [1] Some authors refer to Aut(E/F) as the Galois group for arbitrary extensions EIF and use the corresponding notation, e.g. Jacobson 2009. References • Jacobson, Nathan (2009) [1985], Basic algebra I (Second ed), Dover Publications, ISBN 978-0-486-47189-1 • Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer- Verlag, MR1878556, ISBN 978-0-387-95385-4 External links • Galois Groups (http://www.mathpages.com/home/kmath290/kmath290.htm) at MathPages Grothendieck group In mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid in the best possible way. It takes its name from the more general construction in category theory, introduced by Alexander Grothendieck in his fundamental work of the mid-1950s that resulted in the development of K-theory. The Grothendieck group is denoted by K or K . Universal property In its simplest form, the Grothendieck group of a commutative monoid is the universal way of making that monoid into an abelian group. Let M be a commutative monoid. Its Grothendieck group N should have the following universal property: There exists a monoid homomorphism i:M->N such that for any monoid homomorphism f.M^A from the commutative monoid M to an abelian group A, there is a unique group homomorphism g:N^A such that f=gi- In the language of category theory, the functor that sends a commutative monoid M to its Grothendieck group N is left adjoint to the forgetful functor from the category of abelian groups to the category of commutative monoids. Explicit construction To construct the Grothendieck group of a commutative monoid M, one forms the Cartesian product MxM. The two coordinates are meant to represent a positive part and a negative part: (m, n) is meant to correspond to m - n. Addition is defined coordinate-wise: (rrij, m 2 ) + (re ; , n^ = (m + n } , m 2 + n). Grothendieck group 243 Next we define an equivalence relation on MxM. We say that (m , m ) is equivalent to (n , n ) if, for some element fe of M, m + n + k = m + n + k. It is easy to check that the addition operation is compatible with the equivalence relation. The identity element is now any element of the form (m, m), and the inverse of (m , m ) is (m , m ). In this form, the Grothendieck group is the fundamental construction of K-theory. The group K (M) of a manifold M is defined to be the Grothendieck group of the commutative monoid of all isomorphism classes of vector bundles of finite rank on M with the monoid operation given by direct sum. The Grothendieck group can also be constructed using generators and relations: denoting by (Z(M), + ') the free abelian group generated by the set M, the Grothendieck group is the quotient of Z(M) by the subgroup generated by {x +' y -' (x + y) | x, y € M} . Generalization To apply the Grothendieck group to purely algebraic settings, it is useful to generalize it to the case of an essentially small abelian category. To do this, let j\ be an essentially small abelian category. Let F be the free abelian group generated by isomorphism classes of objects of the category. (This is where the hypothesis of essential smallness is necessary; without it, F would not be a set.) We will impose some relations on F. Call R the subgroup off generated as follows: For each exact sequence 0— »A— »B— >C— >0 in J^ , the element [A] + [C] - [B] is in R. Then the Grothendieck group Kq(jX) is FIR. K of an abelian category has a similar universal property to K of a commutative monoid. We make a preliminary definition: A function x from isomorphism classes of objects of an abelian category j\ to an abelian group A is called additive if, for each exact sequence 0— >A— >B— >C— >0, we have x(A) + x(C) - %{B) = 0. Then, for any additive function %'■ J\. — >A, there is a unique abelian group homomorphism /: KqJ\. — >A such that x factors through/ and the map that takes each object of J^ to the element representing its isomorphism class in Kq(jV\ ■ This universal property makes Kq(jX) the 'universal receiver' of generalized Euler characteristics. In particular, for every bounded complex of objects in j\ > -> -»■ A n -> A n+1 -> > A m_1 -> A™ -> -> -> ■ ■ ■ we have a canonical element [A'] = £(-imi = E(-i)W^)] g *o. In fact the Grothendieck group was originally introduced for the study of Euler characteristics. Splitting principle The relationship between K of a commutative monoid and K of an abelian category comes from the splitting principle. According to the splitting principle, we can always take an exact sequence 0— >A— >5— >C— >0 and find a closely related exact sequence in which the middle term splits, that is, it is the direct sum of the other two terms. Because of this, the Grothendieck group of the commutative monoid of vector bundles on a smooth manifold is the same as the Grothendieck group of the abelian category of vector bundles on that same smooth manifold. K is often defined for a ring or for a ringed space. The usual construction is as follows: For a not necessarily commutative ring R, one lets the abelian category J^ be the category of all finitely generated projective modules over the ring. For a ringed space (X,0 ), one lets the abelian category J^ be the category of all coherent sheaves on X. This makes K into a functor. There is another Grothendieck group of a ring or a ringed space which is sometimes useful. The Grothendieck group G of a ring is the Grothendieck group associated to the category of all finitely generated modules over a ring. Similarly, the Grothendieck group G of a ringed space is the Grothendieck group associated to the category of all Grothendieck group 244 quasicoherent sheaves on the ringed space. G is not a functor, but nevertheless it carries important information. Example The easiest example of the Grothendieck group construction is the construction of the integers from the natural numbers. Let us first check that the natural numbers, f^ , indeed form a monoid. This is easy, the operation should be regular addition and the identity element is zero. We know that addition is associative and indeed fj + n = n + = 7l f° r anv natural number. Now when we use the Grothendieck group construction we obtain the formal differences between natural numbers as elements n — m and we have the equivalence relation n — m ~ n' — m' <-^ n + m' = n' + m- Now define n :— [n — 0], — n :— [0 — n] for all fi (z fsj . This defines the integers ^ ■ In the abelian category of finite dimensional vector spaces over a field & , two vector spaces are isomorphic if and only if they have the same dimension. Thus, for a vector space V the class []/] — r£ ; c ' jm 0'0lin _Ko(Vectfi n ) • Moreover for an exact sequence -> k l -» k m -> k n -c m = I + n, so Thus [V] = dim(V r )[A;l, the Grothendieck group _Ko(Vectfi n )is isomorphic to ^ an d is generated by [k]. Finally for a bounded complex of finite dimensional vector spaces y* , [v*\ = x{v*)[k\ where X is the standard Euler characteristic defined by X(V) = Y^i-lYdimV = ^(-l)MimiT(V*). i i References • Grothendieck group on PlanetMath • Michael F. Atiyah, K-Theory, (Notes taken by D.W.Anderson, Fall 1964), published in 1967, W.A. Benjamin Inc., New York. References [1] http://planetmath.org/?op=getobj&from=ohjects&id=4290 Esquisse d'un Programme 245 Esquisse d'un Programme "Esquisse d'un Programme" is a famous proposal for long-term mathematical research made by the German-born, French mathematician Alexander Grothendieck .He pursued the sequence of logically linked ideas in his important project proposal from 1984 until 1988, but his proposed research continues to date to be of major interest in several branches of advanced mathematics. Grothendieck's vision provides inspiration today for several developments in mathematics such as the extension and generalization of Galois theory, which is currently being extended based on his original proposal. Brief history Submitted in 1984, the Esquisse d'un Programme was a successful proposal submitted by Alexander Grothendieck for a position at the Centre National de la Recherche Scientifique, which Grothendieck held from 1984 till 1988. This proposal was however not formally published until 1997, because the author "could not be found, much less his permission requested". The outlines of dessins d'enfants, a are contained in this manuscript continue to inspire research. permission requested". The outlines of dessins d'enfants, or "children's drawings", and "anabelian geometry", that Abstract of Grothendieck's programme (" Sommaire") 1. The Proposal and enterpise ("Envoi"). 2. "Teichmiiller's Lego-game and the Galois group of Q over Q" ("Un jeu de "Lego-Teichmiiller" et le groupe de Galois de Q sur Q"). 3. Number fields associated with "dessin d'enfants". ("Corps de nombres associes a un dessin d'enfant"). 4. Regular polyhedra over finite fields ("Polyedres reguliers sur les corps finis"). 5. General topology or a 'Moderated Topology' ("Haro sur la topologie dite 'generale', et reflexions heuristiques vers une topologie dite 'moderee"). 6. Differentiable theories and moderated theories ("Theories differentiables" (a la Nash) et "theories moderees"). 7. Pursuing Stacks ("A la Poursuite des Champs"} . 8. Two-dimensional geometry ("Digressions de geometrie bidimensionnelle" . 9. Summary of proposed studies ("Bilan d'une activite enseignante"). 10. Epilogue. Notes Suggested further reading for the interested mathematical reader is provided in the References section. Extensions of Galois's theory for groups: Galois groupoids, categories and functors Galois has developed a powerful, fundamental algebraic theory in mathematics that provides very efficient computations for certain algebraic problems by utilizing the algebraic concept of groups, which is now known as the theory of Galois groups; such computations were not possible before, and also in many cases are much more effective than the 'direct' calculations without using groups . To begin with, Alexander Grothendieck stated in his proposal: "Thus, the group of Galois is realized as the automorphism group of a concrete, pro-finite group which respects certain structures that are essential to this group. " This fundamental, Galois group theory in mathematics has been considerably expanded, at first to groupoids- as proposed in Alexander Grothendieck's Esquisse d' un Programme (EdP)- and now already partially carried out for groupoids; the latter are now further developed beyond groupoids to categories by several groups of mathematicians. Here, we shall focus only on the well-established and fully validated extensions of Galois' theory. Thus, EdP also proposed and anticipated, along previous Alexander Grothendieck's IHES seminars (SGA1 to SGA4) held in the 1960s, the development of even more powerful Esquisse d'un Programme 246 extensions of the original Galois's theory for groups by utilizing categories, functors and natural transformations, as well as further expansion of the manifold of ideas presented in Alexander Grothendieck's Descent Theory. The notion of motive has also been pursued actively. This was developed into the motivic Galois group, Grothendieck ro] topology and Grothendieck category . Such developments were recently extended in algebraic topology via representable functors and the fundamental groupoid functor. Notes [1] Scharlau, Winifred (September 2008), written at Oberwolfach, Germany, "Who is Alexander Grothendieck", Notices of the American Mathematical Society (Providence, RI: American Mathematical Society) 55(8): 930-941, ISSN 1088-9477, OCLC 34550461, http://www. ams.org/notices/200808/tx080800930p.pdf [2] Alexander Grothendieck, 1984. "Esquisse d'un Programme", (1984 manuscript), finally published in Schneps and Lochak (1997, 1), pp.5-48; English transl., ibid., pp. 243-283. MR 99c: 14034 [3] Rehmeyer, Julie (May 9, 2008), "Sensitivity to the Harmony of Things", Science News [4] Schneps and Lochak (1997, 1) p.l [5] http://www.bangor.ac.Uk/r.brown/pstacks.htm [6] Carrier, Pierre (2001), "A mad day's work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry", Bull. Amer. Math. Soc. 38(4): 389-408, <http://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/S0273-0979-01-00913-2.pdf>. An English translation of Cartier (1998) [7] Cartier, Pierre (1998), "La Folle Journee, de Grothendieck a Connes et Kontsevich — Evolution des Notions d'Espace et de Symetrie", Les Relations entre les Mathematiques et la Physique Theorique — Festschrift for the 40th anniversary of the IHES, Institut des Hautes Etudes Scientifiques, pp. 11—19 [8] http://planetmath.org/encyclopedia/GrothendieckCategory.html References Related works by Alexander Grothendieck • Alexander Grothendieck. 1971, Revetements Etales et Groupe Fondamental (SGA1), chapter VI: Categories fibrees et descente, Lecture Notes in Math. 224, Springer- Verlag: Berlin. • Alexander Grothendieck. 1957, Sur quelque point d-algebre homologique. , Tohoku Math. J., 9: 1 19-121. • Alexander Grothendieck and Jean Dieudonne.: 1960, Elements de geometrie algebrique., Publ. Inst, des Hautes Etudes Scientifiques, (IHES), 4. • Alexander Grothendieck et al.,1971. Seminaire de Geometrie Algebrique du Bois-Marie, Vol. 1-7, Berlin: Springer- Verlag. • Alexander Grothendieck. 1962. Seminaires en Geometrie Algebrique du Bois-Marie, Vol. 2 - Cohomologie Locale des Faisceaux Coherents et Theoremes de Lefschetz Locaux et Globaux. , pp. 287. {with an additional contributed expose by Mme. Michele Raynaud). (Typewritten manuscript available in French; see also a brief summary in English References Cited: • Jean-Pierre Serre. 1964. Cohomologie Galoisienne, Springer- Verlag: Berlin. • J. L. Verdier. 1965. Algebre homologiques et Categories derivees. North Holland Publ. Cie). • Alexander Grothendieck. 1957, Sur quelques points d'algebre homologique, Tohoku Mathematics Journal, 9, 119-221. • Alexander Grothendieck et al. Seminaires en Geometrie Algebrique- 4, Tome 1, Expose 1 (or the Appendix to Exposee 1, by "N. Bourbaki' for more detail and a large number of results. AG4 is freely available in French; also available is an extensive Abstract in English. • Alexander Grothendieck, 1984. "Esquisse d'un Programme" (http://people.math.jussieu.fr/~leila/ grothendieckcircle/EsquisseFr.pdf), (1984 manuscript), finally published in "Geometric Galois Actions", L. Schneps, P. Lochak, eds., London Math. Soc. Lecture Notes 242, Cambridge University Press, 1997, pp. 5-48; English transl., ibid., pp. 243-283. MR 99c: 14034 . Esquisse d'un Programme 247 • Alexander Grothendieck, "La longue marche in a travers la theorie de Galois." = "The Long March Towards/Across the Theory of Galois", 1981 manuscript, University of Montpellier preprint series 1996, edited by J. Malgoire. Other related publications • Schneps, Leila (1994), The Grothendieck Theory ofDessins d'Enfants, London Mathematical Society Lecture Note Series, Cambridge University Press. • Schneps, Leila; Lochak, Pierre, eds. (1997), Geometric Galois Actions I: Around Grothendieck' s Esquisse D'un Programme, London Mathematical Society Lecture Note Series, 242, Cambridge University Press, ISBN 978-0-521-59642-8 • Schneps, Leila; Lochak, Pierre, eds. (1997), Geometric Galois Actions II: The Inverse Galois Problem, Moduli Spaces and Mapping Class Groups, London Mathematical Society Lecture Note Series, 243, Cambridge University Press, ISBN 978-0521596411 • Harbater, David; Schneps, Leila (2000), "Fundamental groups of moduli and the Grothendieck— Teichmuller group", Trans. Amer. Math. Soc. 352: 3117-3148, doi:10.1090/S0002-9947-00-02347-3. External links • Fundamental Groupoid Functors (http://planetphysics.org/encyclopedia/QuantumFundamentalGroupoid3. html), Planet Physics. • The best rejected proposal ever (http://www.neverendingbooks.org/index.php/ the-best-rejected-proposal-ever.html), Never Ending Books, Lieven le Bruyn Galois theory In mathematics, more specifically in abstract algebra, Galois theory, named after Evariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood. Originally Galois used permutation groups to describe how the various roots of a given polynomial equation are related to each other. The modern approach to Galois theory, developed by Richard Dedekind, Leopold Kronecker and Emil Artin, among others, involves studying automorphisms of field extensions. Further abstraction of Galois theory is achieved by the theory of Galois connections. Application to classical problems The birth of Galois theory was originally motivated by the following question, whose answer is known as the Abel— Ruffini theorem. "Why is there no formula for the roots of a fifth (or higher) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc)?" Galois theory 248 Galois theory not only provides a beautiful answer to this question, it also explains in detail why it is possible to solve equations of degree four or lower in the above manner, and why their solutions take the form that they do. Further, it gives a conceptually clear, and often practical, means of telling when some particular equation of higher degree can be solved in that manner. Galois theory also gives a clear insight into questions concerning problems in compass and straightedge construction. It gives an elegant characterisation of the ratios of lengths that can be constructed with this method. Using this, it becomes relatively easy to answer such classical problems of geometry as "Which regular polygons are constructible polygons?" "Why is it not possible to trisect every angle?" History Galois theory originated in the study of symmetric functions — the coefficients of a polynomial are (up to sign) the 2 elementary symmetric polynomials in the roots. For instance, (x — a)(x — b) = x — (a + b)x + ab, where 1, a + b and ab are the elementary polynomials of degree 0, 1 and 2 in two variables. This was first formalized by the 16th century French mathematician Francois Viete, in Viete's formulas, for the case of positive real roots. In the opinion of the 18th century British mathematician Charles Hutton, the expression of coefficients of a polynomial in terms of the roots (not only for positive roots) was first understood by the 17th century French mathematician Albert Girard; Hutton writes: ...[Girard was] the first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products. He was the first who discovered the rules for summing the powers of the roots of any equation. In this vein, the discriminant is a symmetric function in the roots which reflects properties of the roots — it is zero if and only if the polynomial has a multiple root, and for quadratic and cubic polynomials it is positive if and only if all roots are real and distinct, and negative if and only if there is a pair of distinct complex conjugate roots. See Discriminant: nature of the roots for details. The cubic was first partly solved by the 15th/16th century Italian mathematician Scipione del Ferro, who did not however publish his results; this method only solved one of three classes, as the others involved taking square roots of negative numbers, and complex numbers were not known at the time. This solution was then rediscovered independently in 1535 by Niccolo Fontana Tartaglia, who shared it with Gerolamo Cardano, asking him to not publish it. Cardano then extended this to the other two cases, using square roots of negatives as intermediate steps; see details at Cardano's method. After the discovery of Ferro's work, he felt that Tartaglia's method was no longer secret, and thus he published his complete solution in his 1545 Ars Magna. His student Lodovico Ferrari solved the quartic polynomial, which solution Cardano also included in Ars Magna. A further step was the 1770 paper Reflexions sur la resolution algebrique des equations by the French-Italian mathematician Joseph Louis Lagrange, in his method of Lagrange resolvents, where he analyzed Cardano and Ferrarri's solution of cubics and quartics by considering them in terms of permutations of the roots, which yielded an auxiliary polynomial of lower degree, providing a unified understanding of the solutions and laying the groundwork for group theory and Galois theory. Crucially, however, he did not consider composition of permutations. Lagrange's method did not extend to quintic equations or higher, because the resolvent had higher degree. The quintic was almost proven to have no general solutions by radicals by Paolo Ruffini in 1799, whose key insight was to use permutation groups, not just a single permutation. His solution contained a gap, which Cauchy considered minor, though this was not patched until the work of Norwegian mathematician Niels Henrik Abel, who published a proof in 1824, thus establishing the Abel— Ruffini theorem. While Ruffini and Abel established that the general quintic could not be solved, some particular quintics can be solved, such as (x - 1) , and the precise criterion by which a given quintic or higher polynomial could be determined Galois theory 249 to be solvable or not was given by Evariste Galois, who showed that whether a polynomial was solvable or not was equivalent to whether or not the permutation group of its roots — in modern terms, its Galois group — had a certain structure — in modern terms, whether or not it was a solvable group. This group was always solvable for polynomials of degree four or less, but not always so for polynomials of degree five and greater, which explains why there is no general solution in higher degree. The permutation group approach to Galois theory Given a polynomial, it may be that some of the roots are connected by various algebraic equations. For example, it 2 3 may be that for two of the roots, say A and B, that A +55 =7. The central idea of Galois theory is to consider those permutations (or rearrangements) of the roots having the property that any algebraic equation satisfied by the roots is still satisfied after the roots have been permuted. An important proviso is that we restrict ourselves to algebraic equations whose coefficients are rational numbers. (One might instead specify a certain field in which the coefficients should lie but, for the simple examples below, we will restrict ourselves to the field of rational numbers.) These permutations together form a permutation group, also called the Galois group of the polynomial (over the rational numbers). To illustrate this point, consider the following examples: First example — a quadratic equation Consider the quadratic equation x 2 -Ax + 1 = 0. By using the quadratic formula, we find that the two roots are ,4 = 2 + ^ B = 2 - >/3. Examples of algebraic equations satisfied by A and B include A + B = 4, and AB = 1. Obviously, in either of these equations, if we exchange A and B, we obtain another true statement. For example, the equation A + B = 4 becomes simply B + A = 4. Furthermore, it is true, but far less obvious, that this holds for every possible algebraic equation with rational coefficients satisfied by the roots A and B; to prove this requires the theory of symmetric polynomials. 2 We conclude that the Galois group of the polynomial x - 4x + 1 consists of two permutations: the identity permutation which leaves A and B untouched, and the transposition permutation which exchanges A and B. It is a cyclic group of order two, and therefore isomorphic to Z/2Z. One might object that A and B are related by yet another algebraic equation, A-B- 2V3 = which does not remain true when A and B are exchanged. However, this equation does not concern us, because it does not have rational coefficients; in particular, _2\/3^ s not ra ti° na l- 2 A similar discussion applies to any quadratic polynomial ax + bx + c, where a, b and c are rational numbers. 2 2 • If the polynomial has only one root, for example x - 4x + 4 = (x—2) , then the Galois group is trivial; that is, it contains only the identity permutation. 2 • If it has two distinct rational roots, for example x - 3x + 2 = (x-2)(x-l), the Galois group is again trivial. • If it has two irrational roots (including the case where the roots are complex), then the Galois group contains two permutations, just as in the above example. Galois theory 250 Second example Consider the polynomial re 4 - lOz 2 + 1, which can also be written as (^ _ 3 )2 _ 24 We wish to describe the Galois group of this polynomial, again over the field of rational numbers. The polynomial has four roots: A = V2 + V3 c = -V2 + V3 There are 24 possible ways to permute these four roots, but not all of these permutations are members of the Galois group. The members of the Galois group must preserve any algebraic equation with rational coefficients involving A, B, C and D. One such equation is A + D = 0. However, since A + C = 2v / 3^0' the permutation (A, B, C, D) -H> (A, B, D, C) is not permitted (because it transforms the valid equation A + D = into the invalid equation A + C = 0). Another equation that the roots satisfy is (A + Bf = 8. This will exclude further permutations, such as (A, B, C, D) -^ (A, C, B, D). Continuing in this way, we find that the only permutations (satisfying both equations simultaneously) remaining are (A, B, C, D) -^ (A, B, C, D) (A, B, C, D) -H> (C, D, A, B) (A, B, C, D) -^ (B, A, D, C) (A, B, C, D) -^ (D, C, B, A), and the Galois group is isomorphic to the Klein four-group. The modern approach by field theory In the modern approach, one starts with a field extension UK (read: L over K), and examines the group of field automorphisms ofLIK (these are mappings a: L — > L with a(x) = x for all x in K). See the article on Galois groups for further explanation and examples. The connection between the two approaches is as follows. The coefficients of the polynomial in question should be chosen from the base field K. The top field L should be the field obtained by adjoining the roots of the polynomial in question to the base field. Any permutation of the roots which respects algebraic equations as described above gives rise to an automorphism of L/K, and vice versa. In the first example above, we were studying the extension Q(V3)/Q, where Q is the field of rational numbers, and Q(V3) is the field obtained from Q by adjoining V3. In the second example, we were studying the extension Galois theory 251 Q(A,B,C,D)/Q. There are several advantages to the modern approach over the permutation group approach. • It permits a far simpler statement of the fundamental theorem of Galois theory. • The use of base fields other than Q is crucial in many areas of mathematics. For example, in algebraic number theory, one often does Galois theory using number fields, finite fields or local fields as the base field. • It allows one to more easily study infinite extensions. Again this is important in algebraic number theory, where for example one often discusses the absolute Galois group of Q, defined to be the Galois group of K/Q where K is an algebraic closure of Q. • It allows for consideration of inseparable extensions. This issue does not arise in the classical framework, since it was always implicitly assumed that arithmetic took place in characteristic zero, but nonzero characteristic arises frequently in number theory and in algebraic geometry. • It removes the rather artificial reliance on chasing roots of polynomials. That is, different polynomials may yield the same extension fields, and the modern approach recognizes the connection between these polynomials. Solvable groups and solution by radicals The notion of a solvable group in group theory allows one to determine whether a polynomial is solvable in the radicals, depending on whether its Galois group has the property of solvability. In essence, each field extension UK corresponds to a factor group in a composition series of the Galois group. If a factor group in the composition series is cyclic of order «, then if the corresponding field extension is an extension of a field containing a primitive root of unity, then it is a radical extension, and the elements of L can then be expressed using the nth root of some element ofK. If all the factor groups in its composition series are cyclic, the Galois group is called solvable, and all of the elements of the corresponding field can be found by repeatedly taking roots, products, and sums of elements from the base field (usually Q). One of the great triumphs of Galois Theory was the proof that for every n > 4, there exist polynomials of degree n which are not solvable by radicals — the Abel— Ruffini theorem. This is due to the fact that for n> A the symmetric group S contains a simple, non-cyclic, normal subgroup. A non-solvable quintic example Van der Waerden cites the polynomial f(x) = x 5 - x- 1- By the rational root theorem it has no rational zeros. Neither does it have linear factors modulo 2 or 3. (1.1673, 0) The polynomial f(x) = X — X — 1 . The lone real root X =1.1673... is algebraic, but not expressible as radicals. The other four roots are complex numbers. / (x)has the factorization (x 2, + x + l)(x 3 + x 2 + l)modulo 2. That means its Galois group modulo 2 is cyclic of order 6. f(x) has no quadratic factor modulo 3. Thus its Galois group modulo 3 has order 5. Galois theory 252 T21 We know that a Galois group modulo a prime is isomorphic to a subgroup of the Galois group over the rationals. A permutation group on 5 objects with operations of orders 6 and 5 must be the symmetric group S&, which must be the Galois group of f(x) ■ This is one of the simplest examples of a non-solvable quintic polynomial. Serge Lang said that Artin was fond of this example. The inverse Galois problem All finite groups do occur as Galois groups. It is easy to construct field extensions with any given finite group as Galois group, as long as one does not also specify the ground field. For that, choose a field K and a finite group G. Cayley's theorem says that G is (up to isomorphism) a subgroup of the symmetric group S on the elements of G. Choose indeterminates {x }, one for each element a of G, and adjoin them to K to get the field F = K({x }). Contained within F is the field L of symmetric rational functions in the {x }. The Galois group of FIL is S, by a basic result of Emil Artin. G acts on F by restriction of action of S. If the fixed field of this action is M, then, by the fundamental theorem of Galois theory, the Galois group of FIM is G. It is an open problem to prove the existence of a field extension of the rational field Q with a given finite group as Galois group. Hilbert played a part in solving the problem for all symmetric and alternating groups. Igor Shafarevich proved that every solvable finite group is the Galois group of some extension of Q. Various people have solved the inverse Galois problem for selected non-abelian simple groups. Existence of solutions has been shown for all but possibly one (Mathieu group M ) of the 26 sporadic simple groups. There is even a polynomial with integral coefficients whose Galois group is the Monster group. Notes [1] (Funkhouser 1930) [2] V.V. Praslov, Polynomials. (2004), Theorem 5.4.5(a) References • Emil Artin (1998). Galois Theory. Dover Publications. ISBN 0-486-62342-4. (Reprinting of second revised edition of 1944, The University of Notre Dame Press). • Jorg Bewersdorff (2006). Galois Theory for Beginners: A Historical Perspective. American Mathematical Society. ISBN 0-8218-3817-2. . • Harold M. Edwards (1984). Galois Theory. Springer- Verlag. ISBN 0-387-90980-X. (Galois' original paper, with extensive background and commentary.) • Funkhouser, H. Gray (1930). "A short account of the history of symmetric functions of roots of equations" (http:/ /jstor.org/stable/2299273). American Mathematical Monthly (The American Mathematical Monthly, Vol. 37, No. 7) 37 (7): 357-365. doi: 10.2307/2299273. • Nathan Jacobson (1985). Basic Algebra I (2nd ed). W.H. Freeman and Company. ISBN 0-7167-1480-9. (Chapter 4 gives an introduction to the field-theoretic approach to Galois theory.) • Janelidze, G; Borceux, Francis (2001). Galois theories. Cambridge University Press. ISBN 978-0-521-80309-0 (This book introduces the reader to the Galois theory of Grothendieck, and some generalisations, leading to Galois groupoids.) • Lang, Serge (1994). Algebraic Number Theory. Berlin, New York: Springer- Verlag. ISBN 978-0-387-94225-4 • M. M. Postnikov (2004). Foundations of Galois Theory. Dover Publications. ISBN 0-486-43518-0. • Ian Stewart (1989). Galois Theory. Chapman and Hall. ISBN 0-412-34550-1. • Volklein, Helmut (1996). Groups as Galois groups: an introduction. Cambridge University Press. ISBN 978-0-521-56280-5 • van der Waerden, Bartel Leendert (1930). Algebra Galois theory 253 • Pop, Florian (2001). "(Some) New Trends in Galois Theory and Arithmetic" (http://www.math.upenn.edu/ -pop/Research/ files-Res/JapanO 1 .pdf) External links Some on-line tutorials on Galois theory appear at: • http://www.math.niu.edu/~beachy/aaol/galois.html • http ://nrich. maths . org/public/vie wer.php ?obj_id= 1 422 • http :// w w w .j milne . org/math/CourseNotes/ft . html Online textbooks in French, German, Italian and English can be found at: • http://www.galois-group.net/ Grothendieck's Galois theory In mathematics, Grothendieck's Galois theory is a highly abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry. It provides, in the classical setting of field theory, an alternative perspective to that of Emil Artin based on linear algebra, which became standard from about the 1930s. The approach of Alexander Grothendieck is concerned with the category-theoretic properties that characterise the categories of finite G-sets for a fixed profinite group G. For example, G might be the group denoted ^ , which is the inverse limit of the cyclic additive groups Z/nZ — or equivalently the completion of the infinite cyclic group Z for the topology of subgroups of finite index. A finite G-set is then a finite set X on which G acts through a quotient finite cyclic group, so that it is specified by giving some permutation of X. In the above example, a connection with classical Galois theory can be seen by regarding •% as the profinite Galois group Gal(F/F) of the algebraic closure F of any finite field F, over F. That is, the automorphisms of F fixing F are described by the inverse limit, as we take larger and larger finite splitting fields over F. The connection with geometry can be seen when we look at covering spaces of the unit disk in the complex plane with the origin removed: the finite covering realised by the z map of the disk, thought of by means of a complex number variable z, corresponds to the subgroup «.Z of the fundamental group of the punctured disk. The theory of Grothendieck, published in SGA1, shows how to reconstruct the category of G-sets from a fibre functor O, which in the geometric setting takes the fibre of a covering above a fixed base point (as a set). In fact there is an isomorphism proved of the type G = Aut(O), the latter being the group of automorphisms (self-natural equivalences) of O. An abstract classification of categories with a functor to the category of sets is given, by means of which one can recognise categories of G-sets for G profinite. To see how this applies to the case of fields, one has to study the tensor product of fields. Later developments in topos theory make this all part of a theory of atomic toposes. Grothendieck's Galois theory 254 References • Grothendieck, A.; et al. (1971). SGA1 Revetements etales et groupe fondamental, 1960—1961'. Lecture Notes in Mathematics 224. Springer Verlag. • Joyal, Andre; Tierney, Myles (1984). An Extension of the Galois Theory of Grothendieck. Memoirs of the American Mathematical Society. Proquest Info & Learning. ISBN 0821823124. • Borceux, F. and Janelidze, G, Cambridge University Press (2001). Galois theories, ISBN 0521803098 (This book introduces the reader to the Galois theory of Grothendieck, and some generalisations, leading to Galois groupoids.) Notes on Grothendieck's Galois Theory http://arxiv.org/abs/math/0009145vl Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group G associated to a field extension LIK acts in a natural way on some abelian groups, for example those constructed directly from L, but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor. The current theory of Galois cohomology came together around 1950, when it was realised that the Galois cohomology of idele class groups in algebraic number theory was one way to formulate class field theory, at the time in the process of ridding itself of connections to L-functions. Galois cohomology makes no assumption that Galois groups are abelian groups, so that this was a non-abelian theory. It was formulated abstractly as a theory of class formations. Two developments of the 1960s turned the position around. Firstly, Galois cohomology appeared as the foundational layer of etale cohomology theory (roughly speaking, the theory as it applies to zero-dimensional schemes). Secondly, non-abelian class field theory was launched as part of the Langlands philosophy, which meant that L-functions were back, with a vengeance. The earliest results identifiable as Galois cohomology had been known long before, in algebraic number theory and the arithmetic of elliptic curves. The normal basis theorem implies that the first cohomology group of the additive group of L will vanish; this is a result on general field extensions, but was known in some form to Richard Dedekind. The corresponding result for the multiplicative group is known as Hilbert's Theorem 90, and was known before 1900. Kummer theory was another such early part of the theory, giving a description of the connecting homomorphism coming from the m-th power map. In fact for a while the multiplicative case of a 1-cocycle for groups that are not necessarily cyclic was formulated as the solubility of Noether's equations, named for Emmy Noether; they appear under this name in Emil Artin's treatment of Galois theory, and may have been folklore in the 1920s. The case of 2-cocycles for the multiplicative group is that of the Brauer group, and the implications seem to have been well known to algebraists of the 1930s. In another direction, that of torsors, these were already implicit in the infinite descent arguments of Fermat for elliptic curves. Numerous direct calculations were done, and the proof of the Mordell— Weil theorem had to proceed by some surrogate of a finiteness proof for a particular H group. The 'twisted' nature of objects over fields that are not algebraically closed, which are not isomorphic but become so over the algebraic closure, was also known in many cases linked to other algebraic groups (such as quadratic forms, simple algebras, Severi— Brauer varieties), in the 1930s, before the general theory arrived. The needs of number theory were in particular expressed by the requirement to have control of a local-global principle for Galois cohomology. This was formulated by means of results in class field theory, such as Hasse's norm theorem. In the case of elliptic curves it led to the key definition of the Tate— Shafarevich group in the Selmer group, Galois cohomology 255 which is the obstruction to the success of a local-global principle. Despite its great importance, for example in the Birch and Swinnerton-Dyer conjecture, it proved very difficult to get any control of it, until results of Karl Rubin gave a way to show in some cases it was finite (a result generally believed, since its conjectural order was predicted by an L-function formula). The other major development of the theory, also involving John Tate was the Tate— Poitou duality result. Technically speaking, G may be a profinite group, in which case the definitions need to be adjusted to allow only continuous cochains. References • Serre, Jean-Pierre (2002), Galois cohomology, Springer Monographs in Mathematics, Berlin, New York: Springer- Verlag, MR1867431, ISBN 978-3-540-42192-4, translation of Cohomologie Galoisienne, Springer- Verlag Lecture Notes 5 (1964). • Milne, James S. (2006), Arithmetic duality theorems [1] (2nd ed.), Charleston, SC: BookSurge, LLC, MR2261462, ISBN 978-1-4196-4274-6 • Neukirch, Jiirgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, 323, Berlin: Springer- Verlag, MR1737196, ISBN 978-3-540-66671-4 References [1] http://www.jmilne.org/math/Books/adt.html Homological algebra Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincare and David Hilbert. The development of homological algebra was closely intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of chain complexes, which can be studied both through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences. From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes. Homological algebra 256 Chain complexes and homology The chain complex is the central notion of homological algebra. It is a sequence (C., d.)of abelian groups and group homomorphisms, with the property that the composition of any two consecutive maps is zero: C. : ■ ■ ■ — > C n+ i — >C n — >C n -i — ->■ ■ ■ ■ , d n o a^+i = 0. The elements of C are called n-chains and the homomorphisms d are called the boundary maps or differentials. The chain groups C may be endowed with extra structure; for example, they may be vector spaces or modules over a fixed ring R. The differentials must preserve the extra structure if it exists; for example, they must be linear maps or homomorphisms of 7?-modules. For notational convenience, restrict attention to abelian groups (more correctly, to the category Ab of abelian groups); a celebrated theorem by Barry Mitchell implies the results will generalize to any abelian category. Every chain complex defines two further sequences of abelian groups, the cycles Z = Ker d and the boundaries B = Im d , where Ker d and Im d denote the kernel and the image of d. Since the composition of two consecutive boundary maps is zero, these groups are embedded into each other as Subgroups of abelian groups are automatically normal; therefore we can define the nth homology group H (C) as the factor group of the n-cycles by the n-boundaries, H n (C) = Z n /B n = Kerd n /Imd n+1 . A chain complex is called acyclic or an exact sequence if all its homology groups are zero. Chain complexes arise in abundance in algebra and algebraic topology. For example, if X is a topological space then the singular chains C (X) are formal linear combinations of continuous maps from the standard n-simplex into X; if K is a simplicial complex then the simplicial chains C (K) are formal linear combinations of the n-simplices of X; if A = FIR is a presentation of an abelian group A by generators and relations, where F is a free abelian group spanned by the generators and R is the subgroup of relations, then letting C (A) = R, C ' (A) = F, and C (A) = for all other n defines a sequence of abelian groups. In all these cases, there are natural differentials d making C into a chain complex, whose homology reflects the structure of the topological space X, the simplicial complex K, or the abelian group A. In the case of topological spaces, we arrive at the notion of singular homology, which plays a fundamental role in investigating the properties of such spaces, for example, manifolds. On a philosophical level, homological algebra teaches us that certain chain complexes associated with algebraic or geometric objects (topological spaces, simplicial complexes, 7?-modules) contain a lot of valuable algebraic information about them, with the homology being only the most readily available part. On a technical level, homological algebra provides the tools for manipulating complexes and extracting this information. Here are two general illustrations. • Two objects X and Y are connected by a map/between them. Homological algebra studies the relation, induced by the map/, between chain complexes associated with X and Y and their homology. This is generalized to the case of several objects and maps connecting them. Phrased in the language of category theory, homological algebra studies the functorial properties of various constructions of chain complexes and of the homology of these complexes. • An object X admits multiple descriptions (for example, as a topological space and as a simplicial complex) or the complex C m (X) is constructed using some 'presentation' of X, which involves non-canonical choices. It is important to know the effect of change in the description of X on chain complexes associated with X. Typically, the complex and its homology H m (C) are functorial with respect to the presentation; and the homology (although not the complex itself) is actually independent of the presentation chosen, thus it is an invariant of X. Homological algebra 257 Functoriality A continuous map of topological spaces gives rise to a homomorphism between their nth homology groups for all n. This basic fact of algebraic topology finds a natural explanation through certain properties of chain complexes. Since it is very common to study several topological spaces simultaneously, in homological algebra one is led to simultaneous consideration of multiple chain complexes. A morphism between two chain complexes, F : C. — ► D u , is a family of homomorphisms of abelian groups C D F :C — > D that commute with the differentials, in the sense that F • d = d • F for all n. A morphism of n n n n -1 n n n chain complexes induces a morphism H u (F)of their homology groups, consisting of the homomorphisms H (F): H (C) — > H (D) for all n. A morphism F is called a quasi-isomorphism if it induces an isomorphism on the nth homology for all n. Many constructions of chain complexes arising in algebra and geometry, including singular homology, have the following functoriality property: if two objects X and Y are connected by a map /, then the associated chain complexes are connected by a morphism F = C(f) from C m (X) to C m (Y), and moreover, the composition g */of maps /:X— >y and g: Y — > Z induces the morphism C(g'f) from C u (X)lo C m (Z) that coincides with the composition C(g) • C(f). It follows that the homology groups H u (C) are functorial as well, so that morphisms between algebraic or topological objects give rise to compatible maps between their homology. The following definition arises from a typical situation in algebra and topology. A triple consisting of three chain complexes L m , M m , iV.and two morphisms between them, f : L m — > M m , g : M m — > iV.,is called an exact triple, or a short exact sequence of complexes, and written as o — > l.-Um.-^n. — ► 0, if for any n, the sequence — > L n ^M n -^N n — > is a short exact sequence of abelian groups. By definition, this means that/ is an injection, g is a surjection, and lm / = Ker g . One of the most basic theorems of homological algebra, sometimes known as the zig-zag lemma, states that, in this case, there is a long exact sequence in homology . . . — ► H^Lf^H^Mf^H^N^H^Lf^^H^M) —>..., where the homology groups of L, M, and N cyclically follow each other, and S are certain homomorphisms determined by / and g, called the connecting homomorphisms. Topological manifestations of this theorem include the Mayer— Vietoris sequence and the long exact sequence for relative homology. Foundational aspects Cohomology theories have been defined for many different objects such as topological spaces, sheaves, groups, rings, Lie algebras, and C*-algebras. The study of modern algebraic geometry would be almost unthinkable without sheaf cohomology. Central to homological algebra is the notion of exact sequence; these can be used to perform actual calculations. A classical tool of homological algebra is that of derived functor; the most basic examples are functors Ext and Tor. With a diverse set of applications in mind, it was natural to try to put the whole subject on a uniform basis. There were several attempts before the subject settled down. An approximate history can be stated as follows: • Cartan-Eilenberg: In their 1956 book "Homological Algebra", these authors used projective and injective module resolutions. • Tohoku': The approach in a celebrated paper by Alexander Grothendieck which appeared in the Second Series of the Tohoku Mathematical Journal in 1957, using the abelian category concept (to include sheaves of abelian groups). Homological algebra 258 • The derived category of Grothendieck and Verdier. Derived categories date back to Verdier's 1967 thesis. They are examples of triangulated categories used in a number of modern theories. These move from computability to generality. The computational sledgehammer par excellence is the spectral sequence; these are essential in the Cartan-Eilenberg and Tohoku approaches where they are needed, for instance, to compute the derived functors of a composition of two functors. Spectral sequences are less essential in the derived category approach, but still play a role whenever concrete computations are necessary. There have been attempts at 'non-commutative' theories which extend first cohomology as torsors (important in Galois cohomology). References • Henri Cartan, Samuel Eilenberg, Homological algebra. With an appendix by David A. Buchsbaum. Reprint of the 1956 original. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1999. xvi+390 pp. ISBN 0-691-04991-2 • Alexander Grothendieck, Sur quelques points d'algebre homologique. Tohoku Math. J. (2) 9, 1957, 1 19--221 • Saunders Mac Lane, Homology. Reprint of the 1975 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. x+422 pp. ISBN 3-540-58662-8 • Peter Hilton; Stammbach, U. A course in homological algebra. Second edition. Graduate Texts in Mathematics, 4. Springer-Verlag, New York, 1997. xii+364 pp. ISBN 0-387-94823-6 • Gelfand, Sergei I.; Yuri Manin, Methods of homological algebra. Translated from Russian 1988 edition. Second edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. xx+372 pp. ISBN 3-540-43583-2 • Gelfand, Sergei I.; Yuri Manin, Homological algebra. Translated from the 1989 Russian original by the authors. Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences {Algebra, V, Encyclopaedia Math. Sci., 38, Springer, Berlin, 1994). Springer-Verlag, Berlin, 1999. iv+222 pp. ISBN 3-540-65378-3 • Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, MR1269324, ISBN 978-0-521-55987-4, OCLC 36131259 Homology theory 259 Homology theory In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces. The general idea To any topological space X and any natural number fc , one can associate a set H^fX), whose elements are called ( ]$ -dimensional) homology classes. There is a well-defined way to add and subtract homology classes, which makes iJ^(J^T)into an abelian group, called the &th homology group of X ■ in heuristic terms, the size and structure of H^fX) gives information about the number of fc -dimensional holes in X ■ F° r example, if X is a figure eight, then it has two holes, which in this context count as being one-dimensional. The corresponding homology group H\{X )can be identified with the group Z © Z°f pairs of integers, with one copy of Z for each hole. While it seems very straightforward to say that X has two holes, it is surprisingly hard to formulate this in a mathematically rigorous way; this is a central purpose of homology theory. For a more intricate example, if yis a Klein bottle then Hi(Y) can be identified with Z © Z/2Z ■ This is not just a sum of copies of Z . so it gives more subtle information than just a count of holes. The formal definition of HAX )can be sketched as follows. The elements of HAX )are one-dimensional cycles, except that two cycles are considered to represent the same element if they are homologous. The simplest kind of one-dimensional cycles are just closed curves in X > which could consist of one or more loops. If a closed curve Cocan be deformed continuously within X t0 another closed curve C\, then Cr^and Ciare homologous and so determine the same element of H\{X\- This captures the main geometric idea, but the full definition is somewhat more complex. For details, see singular homology. There is also a version (called simplicial homology) that works when X is presented as a simplicial complex; this is smaller and easier to understand, but technically less flexible. For example, let J 1 be a torus, as shown on the right. Let (J be the pink curve, and let £) be the red one. For integers n and m , we have another closed curve that goes n times around (J and then m times around £) ; this is denoted by jiQ -\- rnD ■ It can be shown that any closed curve in J 1 is homologous to fiC -\- rnD f° r some n and m , and thus that Hi(T)is again isomorphic to Z © Z ■ Cohomology As well as the homology groups HAX), one can define cohomology groups H k (X)- in the common case where each group H^fX)^ isomorphic to Z rfc f° r some r^ £ N , we just have H k (X) = Hom(H k (X),Z), which is again isomorphic to Z rfc , and H k (X) = Hom(H k (X),Z), so i?fe(J^T)and H k (X) determine each other. In general, the relationship between iJ^(J^T)and H k (X)is on ly a little more complicated, and is controlled by the universal coefficient theorem. The main advantage of cohomology Homology theory 260 over a 3 homology is that it has a natural ring structure: there is a way to multiply an j -dimensional cohomology class by dimensional cohomology class to get an i -\- j -dimensional cohomology class. Applications Notable theorems proved using homology include the following: • The Brouwer fixed point theorem: If fis any continuous map from the ball ^ n to itself, then there is a fixed point a G £J™with f[a) — a- • Invariance of domain: If U is an open subset of J^™ and f '. U — > R™ is an injective continuous map, then V = /(C/)is open and /is a homeomorphism between JJ and \F. • The Hairy ball theorem: any vector field on the 2-sphere (or more generally, the 2Jfc -sphere for any k > 1) vanishes at some point. • The Borsuk— Ulam theorem: any continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. (Two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.) Intersection theory and Poincare duality Let M be a compact oriented manifold of dimension n . The Poincare duality theorem gives a natural isomorphism H^(M) ~ H n _k{M), which we can use to transfer the ring structure from cohomology to homology. For any compact oriented submanifold N C M of dimension d , one can define a so-called fundamental class \N\ <E HAM\ ~ H nd (M) ■ If L is another compact oriented submanifold which meets _/\T transversely, it works out that [£,] [N] = [L Cl N] ■ In many cases the group HAM) will have a basis consisting of fundamental classes of submanifolds, in which case the product rule [£,1 [JV] = [L (~l N] gives a very clear geometric picture of the ring structure. Connection with integration Suppose that X i s an open subset of the complex plane, that f(z)i& a holomorphic function on X > an d that (J is a closed curve in X. ■ There is then a standard way to define the contour integral i> f(z)dz, which is a central idea in complex analysis. One formulation of Cauchy's integral theorem is as follows: if Cr^and Ci^re homologous, then <b f{z)dz= <f> f{z)dz. (Many authors consider only the case where JS^" is simply J Co J C\ connected, in which case every closed curve is homologous to the empty curve and so <t> §{£)dz = 0.) This means that we can make sense of &> f(z)dz when c is merely a homology class, or in other words an element of H\(X\ It is also important that in the case where /(z)is the derivative of another function g(z), we always have i> g\z)dz = 0(even when (J is not homologous to zero). This is the simplest case of a much more general relationship between homology and integration, which is most efficiently formulated in terms of differential forms and de Rham cohomology. To explain this briefly, suppose that X is an open subset of ^ N , or more generally, that X is a manifold. One can then define objects called n -forms on _X" . If _X" is open in JJ3, then the 0-forms are just the scalar fields, the 1 -forms are the vector fields, the 2-forms are the same as the 1 -forms, and the 3-forms are the same as the 0-forms. There is also a kind of differentiation operation called the exterior derivative: if a is an n -form, then the exterior derivative is an in -\- l)-form denoted by d(% . The standard operators div, grad and curl from vector calculus can be seen as special cases of this. There is a procedure for integrating an n -form a over an n -cycle (J to get a number <p a Homology theory 26 1 . It can be shown that <h d/3 = Ofor any (n — l)-form /3 , and that d> a depends only on the homology class of (J , provided Stokes's Theorem and Divergence Theorem can be seen as special cases of this. We say that a is closed if da = > an d exact if a. = d/3 for some j3 . It can be shown that dd/3 is always zero, so that exact forms are always closed. The de Rham cohomology group H^Tf(XY s the quotient of the group of closed forms by the subgroup of exact forms. It follows from the above that there is a well-defined pairing H k {X) X H^ R {X) — > R given by integration. Axiomatics and generalised homology There are various different ways to define cohomology groups (for example singular cohomology, Cech cohomology, Alexander— Spanier cohomology or Sheaf cohomology). These give different answers for some exotic spaces, but there is a large class of spaces on which they all agree. This is most easily understood axiomatically: there is a list of properties known as the Eilenberg— Steenrod axioms, and any two constructions that share those properties will agree at least on all finite CW complexes, for example. One of the axioms is the so-called dimension axiom: if pis a single point, then H n (P) = Ofor all n^O, and Hq (P) = Z ■ We can generalise slightly by allowing an arbitrary abelian group A m dimension zero, but still insisting that the groups in nonzero dimension are trivial. It turns out that there is again an essentially unique system of groups satisfying these axioms, which are denoted by HAX] A)- In the common case where each group Hk(X)is isomorphic to y^for some r^ £ N , we just have H^fX; A) = A Tk ■ In general, the relationship between H^(X)^nd H^fX; A)is only a little more complicated, and is again controlled by the Universal More significantly, 'we can drop the dimension axiom altogether. There are a number of different ways to define groups satisfying all the other axioms, including the following: • The stable homotopy groups irf(X) • Various different flavours of cobordism groups: MOAX\ MSOAX\, MUAX\&n& so on. The last of these (known as complex cobordism) is especially important, because of the link with formal group theory via a theorem of Daniel Quillen. • Various different flavours of K-theory: KO* (X) (real periodic K-theory), kO*(X) (real connective), KU*(X) (complex periodic), kUAX\ (complex connective) and so on. • Brown— Peterson homology, Morava K-theory, Morava E-theory, and other theories defined using the algebra of formal groups. • Various flavours of elliptic homology These are called generalised homology theories; they carry much richer information than ordinary homology, but are often harder to compute. Their study is tightly linked (via the Brown representability theorem) to stable homotopy. Homological algebra and homology of other objects A chain complex consists of groups d (for all { £ 2 ) an d homomorphisms d : C; — > C±—\ satisfying drf = Q . This condition shows that the groups Bi = imagefd : C^+i — > C^)are contained in the groups Zi = kerfd : C^ — > Ci— i) > so one can form the quotient groups Hi = Z^l B^ , which are called the homology groups of the original complex. There is a similar theory of cochain complexes, consisting of groups £"and homomorphisms § ■ (y- y (J i+1 - The simplicial, singular, Cech and Alexander— Spanier groups are all defined by first constructing a chain complex or cochain complex, an then taking its homology. Thus, a substantial part of the work in setting up these groups involves the general theory of chain and cochain complexes, which is known as homological algebra. Homology theory 262 One can also associate (co)chain complexes to a wide variety of other mathematical objects, and then take their (co)homology. For example, there are cohomology modules for groups, Lie algebras and so on. Notes References • Hilton, Peter (1988), "A Brief, Subjective History of Homology and Homotopy Theory in This Century" (http:// www.jstor.org/stable/2689545?origin=JSTOR-pdf), Mathematics Magazine 60 (5): 282-291 Homotopical algebra In mathematics, homotopical algebra is a collection of concepts comprising the nonabelian aspects of homological algebra as well as possibly the abelian aspects as special cases. The homotopical nomenclature stems from the fact that a common approach to such generalizations is via abstract homotopy theory and in particular the theory of closed model categories. This subject has received much attention in recent years due to new foundational work of Voevodsky, Friedlander, Suslin, and others resulting in the A homotopy theory for quasiprojective varieties over a field. Voevodsky has used this new algebraic homotopy theory to prove the Milnor conjecture (for which he was awarded the Fields Medal) and later, in collaboration with M. Rost, the full Bloch-Kato conjecture. References • Goerss, P. G.; Jardine, J. F. (1999), Simplicial Homotopy Theory, Progress in Mathematics, 174, Basel, Boston, Berlin: Birkhauser, ISBN 978-3-7643-6064-1 • Hovey, Mark (1999), Model categories, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1359-1 • Quillen, Daniel (1967), Homotopical Algebra, Berlin, New York: Springer- Verlag, ISBN 978-0-387-03914-5 External links • An abstract for a talk on the proof of the full Bloch-Kato conjecture References [1] http://www.math.neu.edu/bhmn/suslin.html Cohomology theory 263 Cohomology theory In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign 'quantities' to the chains of homology theory. From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century; from the initial idea of homology as a topologically invariant relation on chains, the range of applications of homology and cohomology theories has spread out over geometry and abstract algebra. The terminology tends to mask the fact that in many applications cohomology, a contravariant theory, is more natural than homology. At a basic level this has to do with functions and pullbacks in geometric situations: given spaces X and Y, and some kind of function F on Y, for any mapping f : X — > Y composition with f gives rise to a function F of on X. Cohomology groups often also have a natural product, the cup product, which gives them a ring structure. Because of this feature, cohomology is a stronger invariant than homology, as it can differentiate between certain algebraic objects that homology cannot. Definition For a topological space X, the cohomology group H n (X;G), with coefficents in G, is defined to be the quotient Ker(5)/Im(5) at C n (X;G) in the cochain complex ^C-TH-ipf. G)4-C n {X\ G)<r- . . . ^C°(X; G)«-0. Elements in Ker(5) are cocycles and elements in Im(6) are coboundaries. History Although cohomology is fundamental to modern algebraic topology, its importance was not seen for some 40 years after the development of homology. The concept of dual cell structure, which Henri Poincare used in his proof of his Poincare duality theorem, contained the germ of the idea of cohomology, but this was not seen until later. There were various precursors to cohomology. In the mid-1920s, J.W. Alexander and Solomon Lefschetz founded the intersection theory of cycles on manifolds. On an w-dimensional manifold M, a p-cycle and a g-cycle with nonempty intersection will, if in general position, have intersection a {p+q-n)-cyc\t. This enables us to define a multiplication of homology classes H (M)xH (M)^>H (M). p q p+q-n Alexander had by 1930 defined a first cochain notion, based on a p-cochain on a space X having relevance to the small neighborhoods of the diagonal in X p . In 1931, Georges de Rham related homology and exterior differential forms, proving De Rham's theorem. This result is now understood to be more naturally interpreted in terms of cohomology. In 1934, Lev Pontryagin proved the Pontryagin duality theorem; a result on topological groups. This (in rather special cases) provided an interpretation of Poincare duality and Alexander duality in terms of group characters. At a 1935 conference in Moscow, Andrey Kolmogorov and Alexander both introduced cohomology and tried to construct a cohomology product structure. In 1936 Norman Steenrod published a paper constructing Cech cohomology by dualizing Cech homology. Cohomology theory 264 From 1936 to 1938, Hassler Whitney and Eduard Cech developed the cup product (making cohomology into a graded ring) and cap product, and realized that Poincare duality can be stated in terms of the cap product. Their theory was still limited to finite cell complexes. In 1944, Samuel Eilenberg overcame the technical limitations, and gave the modern definition of singular homology and cohomology. In 1945, Eilenberg and Steenrod stated the axioms defining a homology or cohomology theory. In their 1952 book, Foundations of Algebraic Topology, they proved that the existing homology and cohomology theories did indeed satisfy their axioms. In 1948 Edwin Spanier, building on work of Alexander and Kolmogorov, developed Alexander-Spanier cohomology. Cohomology theories Eilenberg- Steenrod theories A cohomology theory is a family of contravariant functors from the category of pairs of topological spaces and continuous functions (or some subcategory thereof such as the category of CW complexes) to the category of Abelian groups and group homomorphisms that satisfies the Eilenberg-Steenrod axioms. Some cohomology theories in this sense are: • simplicial cohomology • singular cohomology • de Rham cohomology • Cech cohomology • sheaf cohomology. Generalized cohomology theories When one axiom (the dimension axiom) is relaxed, one obtains the idea of generalized cohomology theory or extraordinary cohomology theory; this allows theories based on K-theory and cobordism theory. There are others, coming from stable homotopy theory. In this context, singular homology is referred to as ordinary homology. A generalized cohomology theory is "determined by its values on a point", in the sense that if one has a space given by contractible spaces (homotopy equivalent to a point), glued together along contractible spaces, as in a simplicial complex, then the cohomology of the space is determined by the cohomology of a point and the combinatorics of the patching, and effectively computable. Formally, this is computed by the excision theorem, or equivalently the Mayer— Vietoris sequence. Thus the cohomology of a point is a fundamental calculation for any generalized cohomology theory, though the cohomology of particular spaces is also of interest. Other cohomology theories Theories in a broader sense of cohomology include: Andre— Quillen cohomology BRST cohomology Bonar-Claven cohomology Bounded cohomology Coherent cohomology Crystalline cohomology Cyclic cohomology Deligne cohomology Cohomology theory 265 Dirac cohomology Etale cohomology Flat cohomology Galois cohomology Gel'fand-Fuks cohomology Group cohomology Harrison cohomology Hochschild cohomology Intersection cohomology Lie algebra cohomology Local cohomology Motivic cohomology Non-abelian cohomology Perverse cohomology Quantum cohomology Schur cohomology Spencer cohomology Topological Andre-Quillen cohomology Topological Cyclic cohomology Topological Hochschild cohomology T cohomology Notes [1] Spanier, E. H. (2000) "Book reviews: Foundations of Algebraic Topology" Bulletin of the American Mathematical Society 37(1): pp. 1 14-1 15 [2] http://www. webcitation.org/ query ?url=http://www.geocities.com/jefferywinkler2/ktheory3.html&date=2009- 10-26+00:45:56 References • Hatcher, A. (2001) " Algebraic Topology (http://www.math.cornell.edu/~hatcher/AT/ATpage.html)", Cambridge U press, England: Cambridge, p. 198, ISBN 0-521-79160-X and ISBN 0-521-79540-0 • Hazewinkel, M. (ed.) (1988) Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia" Dordrecht, Netherlands: Reidel, Dordrecht, Netherlands, p. 68, ISBN 1-55608-010-7 • E. Cline, B. Parshall, L. Scott and W. van der Kallen, (1977) "Rational and generic cohomology" Inventiones Mathematicae 39(2): pp. 143-163 • Asadollahi, Javad and Salarian, Shokrollah (2007) "Cohomology theories for complexes" Journal of Pure & Applied Algebra 210(3): pp. 771-787 K-theory 266 K-theory In mathematics, K-theory is a tool used in several disciplines. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It also has some applications in operator algebras. It leads to the construction of families of ^-functors, which contain useful but often hard-to-compute information. In physics, K-theory and in particular twisted K-theory have appeared in Type II string theory where it has been conjectured that they classify D-branes, Ramond— Ramond field strengths and also certain spinors on generalized complex manifolds. For details, see also K-theory (physics). Early history The subject can be said to begin with Alexander Grothendieck (1957), who used it to formulate his Grothendieck— Riemann— Roch theorem. It takes its name from the German "Klasse", meaning "class". Grothendieck needed to work with coherent sheaves on an algebraic variety X. Rather than working directly with the sheaves, he defined a group using (isomorphism classes of) sheaves as generators, subject to a relation that identifies any extension of two sheaves with their sum. The resulting group is called .KYj^Hwhen only locally free sheaves are used, or G(X) when all coherent sheaves. Either of these two constructions is referred to as the Grothendieck group; .KYj^nhas cohomological behavior and £?fJY)has homological behavior. If _X" is a smooth variety, the two groups are the same. If it is a smooth affine variety, then all extensions of locally free sheaves split, so group has an alternative definition. In topology, by applying the same construction to vector bundles, Michael Atiyah and Friedrich Hirzebruch defined K(X)ior a topological space X m 1959, and using the Bott periodicity theorem they made it the basis of an extraordinary cohomology theory. It played a major role in the second proof of the Index Theorem (circa 1962). Furthermore this approach led to a noncommutative K -theory for C*-algebras. Already in 1955, Jean-Pierre Serre had used the analogy of vector bundles with projective modules to formulate Serre's conjecture, which states that projective modules over the ring of polynomials over a field are free modules; this assertion is correct, but was not settled until 20 years later. (Swan's theorem is another aspect of this analogy.) In 1959, Serre formed the Grothendieck group construction for rings, and used it to prove a weak form of the conjecture. This application was one of the beginnings of algebraic K-theory. Developments The other historical origin of algebraic K-theory was the work of Whitehead and others on what later became known as Whitehead torsion. There followed a period in which there were various partial definitions of higher K-theory functors. Finally, two useful and equivalent definitions were given by Daniel Quillen using homotopy theory in 1969 and 1972. A variant was also given by Friedhelm Waldhausen in order to study the algebraic K-theory of spaces, which is related to the study of pseudo-isotopies. Much modern research on higher K-theory is related to algebraic geometry and the study of motivic cohomology. The corresponding constructions involving an auxiliary quadratic form received the general name L-theory. It is a major tool of surgery theory. In string theory the K-theory classification of Ramond— Ramond field strengths and the charges of stable D-branes was first proposed in 1997. K-theory 267 Notes [1] http://arxiv.org/abs/math/0602082 [2] hy Ruben Minasian (http://string.lpthe.jussieu.fr/members.pl?key=7), and Gregory Moore (http://www.physics.rutgers.edu/~gmoore) in K-theory and Ramond— Ramond Charge (http://xxx.lanl.gov/abs/hep-th/9710230). References • Atiyah, Michael Francis (1989), K-theory, Advanced Book Classics (2nd ed.), Addison-Wesley, MR1043170, ISBN 978-0-201-09394-0 (Introductory lectures given at Harvard by Atiyah, published from notes taken by D. W. Anderson. Starts by defining vector bundles, assumes little advanced math.). Swan, R. G. (1968), Algebraic K-Theory, Lecture Notes in Mathematics No. 76, Springer (Introductory lectures given by Swan. Starts with some category theory.). Max Karoubi, K-theory, an introduction (http://www.institut.math.jussieu.fr/~karoubi/KBook.html) (1978) Springer- Verlag Allen Hatcher, Vector Bundles & K-Theory (http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html), (2003) K-theory (http://planetmath.org/?op=getobj&from=objects&id=3338) on PlanetMath Examples of K-theory groups (http://planetmath.org/?op=getobj&from=objects&id=4049) on PlanetMath Algebraic K-theory (http://planetmath.org/?op=getobj&from=objects&id=4117) on PlanetMath Examples of algebraic K-theory groups (http://planetmath.org/?op=getobj&from=objects&id=4118) on PlanetMath Fredholm module (http://planetmath.org/?op=getobj&from=objects&id=3329) on PlanetMath K-homology (http://planetmath.org/?op=getobj&from=objects&id=3330) on PlanetMath Max Karoubi's Page (http://www.institut.math.jussieu.fr/~karoubi/) Algebraic K-theory 268 Algebraic K-theory In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequence K n (R) of functors from rings to abelian groups, for all integers n. For historical reasons, the lower K-groups K and K are thought of in somewhat different terms from the higher algebraic K-groups K for n > 2. Indeed, the lower groups are more accessible, and have more applications, than the higher groups. The theory of the higher K-groups is noticeably deeper, and certainly much harder to compute (even when R is the ring of integers). The group K (R) generalises the construction of the ideal class group of a ring, using projective modules. Its development in the 1960s and 1970s was linked to attempts to solve a conjecture of Serre on projective modules that now is the Quillen-Suslin theorem; numerous other connections with classical algebraic problems were found in this era. Similarly, K (R) is a modification of the group of units in a ring, using elementary matrix theory. The group K (R) is important in topology, especially when R is a group ring, because its quotient the Whitehead group contains the Whitehead torsion used to study problems in simple homotopy theory and surgery theory; the group K (R) also contains other invariants such as the finiteness invariant. Since the 1980s, algebraic K-theory has increasingly had applications to algebraic geometry. For example, motivic cohomology is closely related to algebraic ^-theory. History Alexander Grothendieck discovered K-theory in the mid-1950s as a framework to state his far-reaching generalization of the Riemann-Roch theorem. Within a few years, its topological counterpart was considered by Michael Atiyah and Hirzebruch and is now known as topological K-theory. Applications of K-groups were found from 1960 onwards in surgery theory for manifolds, in particular; and numerous other connections with classical algebraic problems were brought out. A little later a branch of the theory for operator algebras was fruitfully developed, resulting in operator K-theory and KK-theory. It also became clear that K-theory could play a role in algebraic cycle theory in algebraic geometry (Gersten's conjecture): here the higher K-groups become connected with the higher codimension phenomena, which are exactly those that are harder to access. The problem was that the definitions were lacking (or, too many and not obviously consistent). Using work of Robert Steinberg on universal central extensions of classical algebraic groups, John Milnor defined the group K (A) of a ring A as the center, isomorphic to H (E(A),Z), of the universal central extension of the group E(A) of infinite elementary matrices over A. (Definitions below.) There is a natural bilinear pairing from K (A) x K (A) to K (A). In the special case of a field k, with K (k) isomorphic to the multiplicative group GL(l,k), computations of Hideya Matsumoto showed that K (k) is isomorphic to the group generated by K (A) x K (A) modulo an easily described set of relations. Eventually the foundational difficulties were resolved (leaving a deep and difficult theory) by Daniel Quillen, who gave several definitions of K (A) for arbitrary non-negative n, via the +-construction and the Q-construction. Algebraic K-theory 269 Lower K-groups The lower K-groups were discovered first, and given various ad hoc descriptions, which remain useful. Throughout, let A be a ring. The functor K takes a ring A to Grothendieck group of the set of isomorphism classes of its finitely generated projective modules, regarded as a monoid under direct sum. Any ring homomorphism A — > B gives a map K (A) by mapping (the class of) a projective A-module M to M® B, making K a covariant functor. (Projective) modules over a field k are vector spaces and K (k) is isomorphic to Z, by dimension. For A a Dedekind ring, K Q (A) = Pic(A) 8 Z, where Pic(A) is the Picard group of A, and similarly the reduced K-theory is given by K (A) = PicA An algebro-geometric variant of this construction is applied to the category of algebraic varieties; it associates with a given algebraic variety X the Grothendieck' s K-group of the category of locally free sheaves (or coherent sheaves) on X. Given a compact topological space X, the topological K-theory K op (X) of (real) vector bundles over X coincides with K of the ring of continuous real-valued functions on X. K l Hyman Bass provided this definition, which generalizes the group of units of a field: K (A) is the abelianization of the infinite general linear group: K X {A) = GL(A) ab = GL(A)f[GL{A),GL{A)] Here GL(A) = colimGL„(A) is the direct limit of the GL , which embeds in GL as the upper left block matrix, and the commutator subgroup agrees with the group generated by elementary matrices E(^4) = [GL(^4), GL(^4)1, by Whitehead's lemma. Indeed, the group GL(^4)/E(A)was first defined and studied by Whitehead, and is called the Whitehead group of the ring A. As E(>1) < SL(yl), one can also define the special Whitehead group SKi(A) :— SL(A)/E(A) . Commutative rings and fields For A a commutative ring, one can define a determinant det : GL(^4) — ► A* to the group of units of A, which vanishes on E(y4)and thus descends to a map det: KAA\ — ► A* ■ This map splits via the map A*—>GLi(A) — > i^i(A)(unit in the upper left corner), and hence is onto, and has the special Whitehead group as kernel, yielding the split short exact sequence: 1 -> SK^A) -»■ K X {A) -> A* ->■ 1, which is a quotient of the usual split short exact sequence defining the special linear group, namely 1 -> SL(A) -> GL(A) -> A* -)- 1. Thus, since the groups in question are abelian, KA A) splits as the direct sum of the group of units and the special Whitehead group: K^A) fti A* © SK^A). When A is a Euclidean domain (e.g. a field, or the integers) SK (A) vanishes, and the determinant map is an isomorphism. In particular, det : K\ (F) ^F* ■ This is false in general for PIDs, thus providing one of the rare mathematical features of Euclidean domains that do not generalize to all PIDs. An explicit PID A such that SK (A) is Algebraic K-theory 270 nonzero was given by Grayson in 1981. A hard theorem of Bass, Milnor, and Serre shows SK (A) vanishes when A is the ring of S-integers in any global field. For a non-commutative ring, the determinant cannot be defined, but the map GL(.A) — ► KAA) generalizes the determinant. K 2 John Milnor found the right definition of K : it is the center of the Steinberg group St(.A) of A. It can also be defined as the kernel of the map ip: St(4) -> GL(A), or as the Schur multiplier of the group of elementary matrices. For a field k one has K 2 (k) = k x ® z k x /(a ® (1 - a) | a ^ 0, 1). Milnor /^-theory The above expression for K of a field k led Milnor to the following definition of "higher" ^-groups by K™{k):=T*k*/{a®(l-a)), thus as graded parts of a quotient of the tensor algebra of the multiplicative group k by the two-sided ideal, generated by the a ® (1 — a) for a * 0,1. For n = 0,1,2 these coincide with those below, but for nU3 they differ in general. For example, we have K (F„) = Ofor nU3. Milnor K-theory modulo 2 is related to etale (or Galois) cohomology of the field by the Milnor conjecture, proven by Voevodsky. Higher /^-theory The master, definitive definitions of ^-theory were given by Daniel Quillen, after an extended period in which uncertainty had reigned. The +-construction One possible definition of higher algebraic ^-theory of rings was given by Quillen K n (R)=7r n (BGL{R)+), Here 7T n is a homotopy group, GL(R) is the direct limit of the general linear groups over R for the size of the matrix tending to infinity, B is the classifying space construction of homotopy theory, and the is Quillen's plus construction. This definition only holds for n>0 so one often defines the higher algebraic ^-theory via K n (R) = K n {BGL{R) + x K Q {R)) Bi for n=0. Since BGL(R) is path connected and K (R) discrete, this definition doesn't differ in higher degrees and also holds Algebraic K-theory 27 1 The Q-construction The Q-construction gives the same results as the +-construction, but it applies in more general situations. Moreover, the definition is more direct in the sense that the ^-groups, defined via the Q-construction are functorial by definition. This fact is not automatic in the +-construction. Suppose P is an exact category; associated toPa new category QP is defined, objects of which are those of P and morphisms from M' to M" are isomorphism classes of diagrams M' < — N — ► M", where the first arrow is an admissible epimorphism and the second arrow is an admissible monomorphism. The i-th £-group of P is then defined as K i (P) = K i+1 (BQP,0) with a fixed zero-object 0, where BQ is the classifying space of Q, which is defined to be the geometric realisation of the nerve of Q. This definition coincides with the above definitions of K , K and K . The ^-groups KAA)of the ring A are then the ^-groups KAPj^) where P^ is the category of finitely generated projective A-modules. More generally, for a scheme X, the higher ^-groups of X are by definition the ^-groups of (the exact category of) locally free coherent sheaves on X. The following variant of this is also used: instead of finitely generated projective (=locally free) modules, take finitely generated modules. The resulting ^-groups are usually called G-groups, or higher G-theory. When A is a noetherian regular ring, then G- and ^-theory coincide. Indeed, the global dimension of regular local rings is finite, i.e. any finitely generated module has a finite projective resolution, so the canonical map K — » G is surjective. It is also injective, as can be shown. This isomorphism extends to the higher ^-groups, too. The S-construction A third construction of ^-theory groups is the S-construction, due to Waldhausen. It applies to categories with cofibrations (also called Waldhausen categories). This is a more general concept than exact categories. Examples While the Quillen algebraic K-theory has provided deep insight into various aspects of algebraic geometry and topology, the K-groups have proved particularly difficult to compute except in a few isolated but interesting cases. Algebraic K-groups of finite fields The first and one of the most important calculations of the higher algebraic K-groups of a ring were made by Quillen himself for the case of finite fields: Theorem. Let F be a finite field with q elements. Then K (F) = Z , K 2i (F) = for i y£ 0, and K 2i - 1 {F) = /V-l for 2 = 1,2,... where /V denotes the cyclic group with r elements. Algebraic K-theory 272 Algebraic K-groups of rings of integers Quillen proved that if A is the ring of algebraic integers in an algebraic number field F (a finite extension of the rationals), then the algebraic K-groups of A are finitely generated. Borel used this to calculate K.(A) and K.(F ) modulo torsion. For example, for the integers Z, Borel proved that (modulo torsion) -ftTj(Z) = Ofor positive i unless { — 4jfc + lwith k positive and (modulo torsion) K ik+ l(Z) — Z for positive k. The torsion subgroups of K (Z), and the orders of the finite groups K (Z) have recently been determined, but whether the latter groups are cyclic, and whether the groups K (Z) vanish depends upon Vandiver's conjecture about the class groups of cyclotomic integers. Applications and open questions Algebraic J{ -groups are used in conjectures on special values of L-functions and the formulation of an non-commutative main conjecture of Iwasawa theory and in construction of higher regulators. Another fundamental conjecture due to Hyman Bass (Bass conjecture) says that all G-groups G(A) (that is to say, ^-groups of the category of finitely generated A-modules) are finitely generated when A is a finitely generated T71 Z-algebra. References [1] Karoubi, Max (2008), K-Theory: an Introduction, Classics in mathematics, Berlin, New York: Springer- Verlag, ISBN 978-3-540-79889-7, see Theorem 1.6.18 [2] J.H.C. Whitehead, Simple homotopy types Amer. J. Math. , 72 (1950) pp. 1-57 [3] Not to be confused with the Whitehead group of a group. [4] (Weibel 2005), cf. Lemma 1.8 [5] Voevodsky, Vladimir (2003), "Motivic cohomology with Z/2-coefficients", Institut des Hautes Etudes Scientiflques. Publications Mathematiques 98 (98): 59-104, doi:10.1007/sl0240-003-0010-6, MR2031199, ISSN 0073-8301 [6] Waldhausen, Friedhelm (1985), Algebraic 'K-theory of spaces, Lecture Notes in Mathematics, 1126, Berlin, New York: Springer- Verlag, pp. 318—419, doi:10.1007/BFb0074449, MR802796. See also Lecture IV and the references in (Friedlander & Weibel 1999) [7] (Friedlander & Weibel 1999), Lecture VI • Friedlander, Eric M.; Weibel, Charles W. (1999), An overview of algebraic K-theory, World Sci. Publ., River Edge, NJ, pp. 1-119, MR17 15873 • Milnor, John Willard (1969 1970), 'Algebraic ^-theory and quadratic forms", Inventiones Mathematicae 9: 318-344, doi:10.1007/BF01425486, MR0260844, ISSN 0020-9910 • Milnor, John Willard (1971), Introduction to algebraic K-theory, Princeton, NJ: Princeton University Press, MR0349811 (lower K-groups) • Quillen, Daniel (1975), "Higher algebraic K-theory", Proceedings of the International Congress of Mathematicians (Vancouver, B. C, 1974), Vol. 1, Montreal, Quebec: Canad. Math. Congress, pp. 171—176, MR0422392 (Quillen's Q-construction) • Quillen, Daniel (1974), "Higher K-theory for categories with exact sequences", New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), London Math. Soc. Lecture Note Ser., 11, Cambridge University Press, pp. 95—103, MR0335604 (relation of Q-construction to +-construction) • Seiler, Wolfgang (1988), "X-Rings and Adams Operations in Algebraic K-Theory", in Rapoport, M.; Schneider, P.; Schappacher, N, Beilinson's Conjectures on Special Values of L-F unctions, Boston, MA: Academic Press, ISBN 978-0-12-581 120-0 • Weibel, Charles (2005), "Algebraic K-theory of rings of integers in local and global fields" (http://www.math. uiuc.edu/K-theory/0691/KZsurvey.pdf), Handbook of K-theory, Berlin, New York: Springer- Verlag, pp. 139-190, MR2181823 (survey article) Algebraic K- theory 273 External links • C. Weibel " The K-book: An introduction to algebraic K-theory (http://www.math.rutgers.edu/~weibel/ Kbook.html)" Topological K-theory In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on general topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early work on topological K-theory is due to Michael Atiyah and Friedrich Hirzebruch. Definitions Let X be a compact Hausdorff space and £ = JJ or fc = C • Then K^fX)^ the Grothendieck group of the commutative monoid whose elements are the isomorphism classes of finite dimensional £ -vector bundles on X with the operation [E] © [F] = [E@F] for vector bundles E, F. Usually, K^fX) is denoted KO(X)in real case and KU(X)in the complex case. More explicitly, stable equivalence, the equivalence relation on bundles E and F on X of defining the same element in K{X), occurs when there is a trivial bundle G, so that E®G = F®G- Under the tensor product of vector bundles K{X) then becomes a commutative ring. The rank of a vector bundle carries over to the K-group. Define the homomorphism K{X) -> H°(X, Z) where H°(X Z)is the 0-group of Cech cohomology which is equal to the group of locally constant functions with values in ^ • If X has a distinguished basepoint x , then the reduced K-group (cf. reduced homology) satisfies K{X) = K(X) © K{{x }) and is defined as either the kernel of K(X) — > K ({xo})( w here {xq\ — > J^" is basepoint inclusion) or the cokernel of K(\xq\) — > iif(JT)(where X — ► {^0} * s me cons tant map). When X is a connected space, K(X) = Kei(K(X) -> H°(X, Z) = Z) • The definition of the functor K extends to the category of pairs of compact spaces (in this category, an object is a pair (X,Y), where _X" is compact and Y d X^ s closed, a morphism between (X,Y)and (X ', Y"')is a continuous map f ; X — > X' such that f(Y) C Y') K(X,Y):=K(X/Y). The reduced K-group is given by xq = {Y\ ■ The definition K£(X,Y) = kc(^{X/Y)) gives the sequence of K-groups for ji ^ % , where S denotes the reduced suspension. Topological K-theory 274 Properties • K n is a contravariant functor. • The classifying space of fr is BO k (BO, in real case; BU in complex case), i.e. Ku(X) = \X BO k ]. • The classifying space of Jf is Z X BO k ( Z with discrete topology), i.e. K k (X) ^[I,Zx BO k ] . • There is a natural ring homomorphism K*(X) — > H 2 *(X Q) . the Chern character, such that K*(X) <g> O -> # 2 *(X, Q)is an isomorphism. • Topological K-theory can be generalized vastly to a functor on C*-algebras, see operator K-theory and KK- theory. Bott periodicity The phenomenon of periodicity named for Raoul Bott (see Bott periodicity theorem) can be formulated this way: • K(X X S 2 ) = K(X) ® K(S 2 ),and K(S 2 ) = Z[H]/(H - l) 2 ; where H is the class of the tautological bundle on the g 2 — C_p 1 > i.e. the Riemann sphere as complex projective line. • k n+2 (x) = k n (x). • fi 2 BU ~ BU x Z. In real K-theory there is a similar periodicity, but modulo 8. Notes References • M. Karoubi, K-theory, an introduction (http://www.institut.math.jussieu.fr/~karoubi/KBook.html), 1978 - Berlin; New York: Springer- Verlag • M.F. Atiyah, D.W. Anderson K-Theory 1967 - New York, WA Benjamin • A. Hatcher Vector Bundles & K-Theory (http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html) (still unfinished) 275 Category Theories Category theory Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows (also called morphisms, although this term also has a specific, non category-theoretical sense), where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories. The most accessible example of a category is the Category of sets, where the objects are sets and the arrows are functions from one set to another. However it is important to note that the objects of a category need not be sets nor the arrows functions; any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category, and all the results of category theory will apply to it. One of the simplest examples of a category (which is a very important concept in topology) is that of groupoid, defined as a category whose arrows or morphisms are all invertible. Categories now appear in most branches of mathematics, some areas of theoretical computer science where they correspond to types, and mathematical physics where they can be used to describe vector spaces. Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942—45, in connection with algebraic topology. Category theory has several faces known not just to specialists, but to other mathematicians. A term dating from the 1940s, "general abstract nonsense", refers to its high level of abstraction, compared to more classical branches of mathematics. Homological algebra is category theory in its aspect of organising and suggesting manipulations in abstract algebra. Diagram chasing is a visual method of arguing with abstract "arrows" joined in diagrams. Note that arrows between categories are called functors, subject to specific defining commutativity conditions; moreover, categorical diagrams and sequences can be defined as functors (viz. Mitchell, 1965). An arrow between two functors is a natural transformation when it is subject to certain naturality or commutativity conditions. Functors and natural transformations ('naturality') are the key concepts in category theory . Topos theory is a form of abstract sheaf theory, with geometric origins, and leads to ideas such as pointless topology. A topos can also be considered as a specific type of category with two additional topos axioms. Category theory 276 Background The study of categories is an attempt to axiomatically capture what is commonly found in various classes of related mathematical structures by relating them to the structure-preserving functions between them. A systematic study of category theory then allows us to prove general results about any of these types of mathematical structures from the axioms of a category. Consider the following example. The class Grp of groups consists of all objects having a "group structure". One can proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it is immediately proved from the axioms that the identity element of a group is unique. Instead of focusing merely on the individual objects (e.g., groups) possessing a given structure, category theory emphasizes the morphisms — the structure-preserving mappings — between these objects; by studying these morphisms, we are able to learn more about the structure of the objects. In the case of groups, the morphisms are the group homo morphisms. A group homomorphism between two groups "preserves the group structure" in a precise sense — it is a "process" taking one group to another, in a way that carries along information about the structure of the first group into the second group. The study of group homomorphisms then provides a tool for studying general properties of groups and consequences of the group axioms. A similar type of investigation occurs in many mathematical theories, such as the study of continuous maps (morphisms) between topological spaces in topology (the associated category is called Top), and the study of smooth functions (morphisms) in manifold theory. If one axiomatizes relations instead of functions, one obtains the theory of allegories. Functors Abstracting again, a category is itself a type of mathematical structure, so we can look for "processes" which preserve this structure in some sense; such a process is called a functor. A functor associates to every object of one category an object of another category, and to every morphism in the first category a morphism in the second. In fact, what we have done is define a category of categories and functors — the objects are categories, and the morphisms (between categories) are functors. By studying categories and functors, we are not just studying a class of mathematical structures and the morphisms between them; we are studying the relationships between various classes of mathematical structures. This is a fundamental idea, which first surfaced in algebraic topology. Difficult topological questions can be translated into algebraic questions which are often easier to solve. Basic constructions, such as the fundamental group or fundamental groupoid of a topological space, can be expressed as fundamental functors to the category of groupoids in this way, and the concept is pervasive in algebra and its applications. Natural transformation Abstracting yet again, constructions are often "naturally related" — a vague notion, at first sight. This leads to the clarifying concept of natural transformation, a way to "map" one functor to another. Many important constructions in mathematics can be studied in this context. "Naturality" is a principle, like general covariance in physics, that cuts deeper than is initially apparent. Historical notes In 1942—45, Samuel Eilenberg and Saunders Mac Lane introduced categories, functors, and natural transformations as part of their work in topology, especially algebraic topology. Their work was an important part of the transition from intuitive and geometric homology to axiomatic homology theory. Eilenberg and Mac Lane later wrote that their goal was to understand natural transformations; in order to do that, functors had to be defined, which required categories. Category theory 277 Stanislaw Ulam, and some writing on his behalf, have claimed that related ideas were current in the late 1930s in Poland. Eilenberg was Polish, and studied mathematics in Poland in the 1930s. Category theory is also, in some sense, a continuation of the work of Emmy Noether (one of Mac Lane's teachers) in formalizing abstract processes; Noether realized that in order to understand a type of mathematical structure, one needs to understand the processes preserving that structure. In order to achieve this understanding, Eilenberg and Mac Lane proposed an axiomatic formalization of the relation between structures and the processes preserving them. The subsequent development of category theory was powered first by the computational needs of homological algebra, and later by the axiomatic needs of algebraic geometry, the field most resistant to being grounded in either axiomatic set theory or the Russell-Whitehead view of united foundations. General category theory, an extension of universal algebra having many new features allowing for semantic flexibility and higher-order logic, came later; it is now applied throughout mathematics. Certain categories called topoi (singular topos) can even serve as an alternative to axiomatic set theory as a foundation of mathematics. These foundational applications of category theory have been worked out in fair detail as a basis for, and justification of, constructive mathematics. More recent efforts to introduce undergraduates to categories as a foundation for mathematics include Lawvere and Rosebrugh (2003) and Lawvere and Schanuel (1997). Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with applications in functional programming and domain theory, where a cartesian closed category is taken as a non-syntactic description of a lambda calculus. At the very least, category theoretic language clarifies what exactly these related areas have in common (in some abstract sense). Categories, objects, and morphisms A category C consists of the following three mathematical entities: • A class ob(C), whose elements are called objects; • A class hom(C), whose elements are called morphisms or maps or arrows. Each morphism/has a unique source object a and target object b. We write/: a — > b, and we say "/is a morphism from a to b". We write hom(a, b) (or Hom(a, b), or hom {a, b), or Mor(a, b), or C(a, b)) to denote the hom-class of all morphisms from a to b. • A binary operation o , called composition of morphisms, such that for any three objects a, b, and c, we have hom(a, b) x hom(Z>, c) — » hom(a, c). The composition of/ a — > b and g: b — » c is written as g o for gf , governed by two axioms: • Associativity: If/: a — > b, g : b — > c and h : c — > d then h o (g o /) = (hog)of, and • Identity: For every object x, there exists a morphism 1 : x — > x called the identity morphism for x, such that for every morphism/: a — > &, we have lj,o/ = / = /ol a . From these axioms, it can be proved that there is exactly one identity morphism for every object. Some authors deviate from the definition just given by identifying each object with its identity morphism. Relations among morphisms (such as/g = h) are often depicted using commutative diagrams, with "points" (corners) representing objects and "arrows" representing morphisms. Category theory 278 Properties of morphisms Some morphisms have important properties. A morphism/: a — » b is: • a monomorphism (or monk) if fag =fag implies g = g for all morphisms g , g : x — > a. • an epimorphism (or epic) if g o/= go/ implies g = g ? for all morphisms g , g : b — > x. • an isomorphism if there exists a morphism g : Z? — > a with/og =1 and go/= 1 . • an endomorphism if a = b. end(a) denotes the class of endomorphisms of a. • an automorphism iff is both an endomorphism and an isomorphism, aut(a) denotes the class of automorphisms of a. Functors Functors are structure-preserving maps between categories. They can be thought of as morphisms in the category of all (small) categories. A (covariant) functor F from a category C to a category D, written F:C — > D, consists of: • for each object x in C, an object F(x) in D; and • for each morphism/: x — > y in C, a morphism F(f) : F(x) — > F(y), such that the following two properties hold: • For every object x in C, F{\ ) = 1 „, • J x F(x) • For all morphisms/: x — > y and g : y — > z, i^(j o /) = Fig) o F(f). A contravariant functor F: C — > D, is like a covariant functor, except that it "turns morphisms around" ("reverses all the arrows"). More specifically, every morphism/: x — » y in C must be assigned to a morphism F(f) : F(y) — > F(x) in D. In other words, a contravariant functor is a covariant functor from the opposite category C op to D. Natural transformations and isomorphisms A natural transformation is a relation between two functors. Functors often describe "natural constructions" and natural transformations then describe "natural homomorphisms" between two such constructions. Sometimes two quite different constructions yield "the same" result; this is expressed by a natural isomorphism between the two functors. If F and G are (covariant) functors between the categories C and D, then a natural transformation r| from F to G associates to every object xinCa morphism r| : F(x) — > G(x) in D such that for every morphism/: x — » y in C, we have 1] o F(f) = G(f) o 1] ; this means that the following diagram is commutative: y x F(f) F{X) ^-+ F{Y) Vx T)y The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G such that r| is an isomorphism for every object x in C. Category theory 279 Universal constructions, limits, and colimits Using the language of category theory, many areas of mathematical study can be cast into appropriate categories, such as the categories of all sets, groups, topologies, and so on. These categories surely have some objects that are "special" in a certain way, such as the empty set or the product of two topologies, yet in the definition of a category, objects are considered to be atomic, i.e., we do not know whether an object A is a set, a topology, or any other abstract concept — hence, the challenge is to define special objects without referring to the internal structure of those objects. But how can we define the empty set without referring to elements, or the product topology without referring to open sets? The solution is to characterize these objects in terms of their relations to other objects, as given by the morphisms of the respective categories. Thus, the task is to find universal properties that uniquely determine the objects of interest. Indeed, it turns out that numerous important constructions can be described in a purely categorical way. The central concept which is needed for this purpose is called categorical limit, and can be dualized to yield the notion of a colimit. Equivalent categories It is a natural question to ask: under which conditions can two categories be considered to be "essentially the same", in the sense that theorems about one category can readily be transformed into theorems about the other category? The major tool one employs to describe such a situation is called equivalence of categories, which is given by appropriate functors between two categories. Categorical equivalence has found numerous applications in mathematics. Further concepts and results The definitions of categories and functors provide only the very basics of categorical algebra; additional important topics are listed below. Although there are strong interrelations between all of these topics, the given order can be considered as a guideline for further reading. c • The functor category D has as objects the functors from C to D and as morphisms the natural transformations of such functors. The Yoneda lemma is one of the most famous basic results of category theory; it describes representable functors in functor categories. • Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by "reversing all the arrows". If one statement is true in a category C then its dual will be true in the dual category C op . This duality, which is transparent at the level of category theory, is often obscured in applications and can lead to surprising relationships. • Adjoint functors: A functor can be left (or right) adjoint to another functor that maps in the opposite direction. Such a pair of adjoint functors typically arises from a construction defined by a universal property; this can be seen as a more abstract and powerful view on universal properties. Category theory 280 Higher-dimensional categories Many of the above concepts, especially equivalence of categories, adjoint functor pairs, and functor categories, can be situated into the context of higher-dimensional categories. Briefly, if we consider a morphism between two objects as a "process taking us from one object to another", then higher-dimensional categories allow us to profitably generalize this by considering "higher-dimensional processes". For example, a (strict) 2-category is a category together with "morphisms between morphisms", i.e., processes which allow us to transform one morphism into another. We can then "compose" these "bimorphisms" both horizontally and vertically, and we require a 2-dimensional "exchange law" to hold, relating the two composition laws. In this context, the standard example is Cat, the 2-category of all (small) categories, and in this example, bimorphisms of morphisms are simply natural transformations of morphisms in the usual sense. Another basic example is to consider a 2-category with a single object; these are essentially monoidal categories. Bicategories are a weaker notion of 2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative "up to" an isomorphism. This process can be extended for all natural numbers n, and these are called n-categories. There is even a notion of m-category corresponding to the ordinal number a>. Higher-dimensional categories are part of the broader mathematical field of higher-dimensional algebra, a concept introduced by Ronald Brown. For a conversational introduction to these ideas, see John Baez, 'A Tale of «-categories' (1996). See also Domain theory Enriched category theory Glossary of category theory Higher category theory Higher-dimensional algebra Important publications in category theory Timeline of category theory and related mathematics Notes [1] Categories for the Working Mathematician, 2nd Edition, p 18: "As Eilenberg-Mac Lane first observed, 'category' has been defined in order to be able to define 'functor' and 'functor' has been defined in order to be able to define 'natural transformation' ". [2] http://planetphysics.org/encyclopedia/FundamentalGroupoidFunctor.html [3] Some authors compose in the opposite order, writing/g or J O g for g O J . Computer scientists using category theory very commonly write fig for g O J [4] Note that a morphism that is both epic and monic is not necessarily an isomorphism! For example, in the category consisting of two objects A and B, the identity morphisms, and a single morphism/from A to B, /is both epic and monic but is not an isomorphism. [5] http://math.ucr.edu/home/baez/week73.html Category theory 28 1 References • Adamek, Jiff; Herrlich, Horst; Strecker, George E. (1990). Abstract and concrete categories (http://katmat.math. uni-bremen.de/acc/acc.htm). John Wiley & Sons. ISBN 0-471-60922-6. Awodey, Steve (2006). Category Theory (Oxford Logic Guides 49). Oxford University Press. 2nd edition, 2010. Barr, Michael; Wells, Charles (1999). "Category Theory Lecture Notes" (http://folli.loria.fr/cds/1999/library/ pdf/barrwells.pdf). Retrieved 11 December 2009-12-11. Based on their book Category Theory for Computing Science. Barr, Michael; Wells, Charles (2002). Toposes, triples and theories (http://www.cwru.edu/artsci/math/wells/ pub/ttt.html). Revised and corrected translation of Grundlehren der mathematischen Wissenschaften. Springer- Verlag. 1983. Borceux, Francis (1994). Handbook of categorical algebra (Encyclopedia of Mathematics and its Applications 50-52). Cambridge Univ. Press. I. Bucur, A. Deleanu (1968). "Introduction to the theory of categories and functors", Wiley. Freyd, Peter J. (1964). Abelian Categories (http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html). New York: Harper and Row. Freyd, Peter J.; Scedrov, Andre (1990). Categories, allegories. North Holland Mathematical Library 39. North Holland. Goldblatt, R. (1984). Topoi: The Categorial Analysis of Logic (http://dlxs2. library. Cornell. edu/cgi/t/text/ text-idx?c=math;cc=math;view=toc;subview=short;idno=Gold010). Hatcher, William S. (1982). "8". In Pergamon. The Logical Foundations of Mathematics (2nd ed.). Herrlich, Horst; Strecker, George E. (2007), Category Theory (3rd ed.), Heldermann Verlag Berlin, ISBN 978-3-88538-001-6. Masaki Kashiwara, Pierre Schapira, "Categories and Sheaves", Grundlehren der Mathematischen Wissenschaften 332, Springer (2000) Lawvere, William; Rosebrugh, Robert (2003). Sets for mathematics. Cambridge University Press. Lawvere, William; Schanuel, Steve (1997). Conceptual mathematics: a first introduction to categories. Cambridge University Press. Leinster, Tom (2004). Higher operads, higher categories (http://www.maths.gla.ac.uk/~tl/book.html). London Math. Society Lecture Note Series 298. Cambridge University Press. Lurie, Jacob (2009). Higher topos theory (http://arxiv.org/abs/math.CT/0608040). Annals of Mathematics Studies 170. Princeton University Press. Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 (2nd ed.). Springer- Verlag. ISBN 0-387-98403-8. Mac Lane, Saunders; Birkhoff, Garrett (1999) [1967]. Algebra (2nd ed.). Chelsea. ISBN 0-8218-1646-2. Martini, A.; Ehrig, H.; Nunes, D. (1996). "Elements of basic category theory" (http://citeseer.ist.psu.edu/ martini96element.html). Technical Report (Technical University Berlin) 96 (5). May, Peter (1999). A Concise Course in Algebraic Topology. University of Chicago Press. ISBN 0-226-51183-9. Pedicchio, Maria Cristina; Tholen, Walter (2004). Categorical foundations . Encyclopedia of Mathematics and its Applications 97. Cambridge University Press. Pierce, Benjamin (1991). Basic Category Theory for Computer Scientists. MIT Press. Schalk, A.; Simmons, H. (2005). An introduction to Category Theory in four easy movements (http://www.es. man.ac.uk/~hsimmons/BOOKS/CatTheory.pdf). Notes for a course offered as part of the MSc. in Mathematical Logic, Manchester University. Simpson, Carlos. Homotopy theory of higher categories (http://arxiv.org/abs/1001.4071)., draft of a book. Taylor, Paul (1999). Practical Foundations of Mathematics. Cambridge University Press. Turi, Daniele (1996—2001). "Category Theory Lecture Notes" (http://www.dcs.ed.ac.uk/home/dt/CT/ categories.pdf). Retrieved 1 1 December 2009. Based on Mac Lane (1998). Category theory 282 External links • Chris Hillman, A Categorical Primer (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.24.3264& rep=repl&type=pdf), formal introduction to category theory. • J. Adamek, H. Herrlich, G. Stecker, Abstract and Concrete Categories-The Joy of Cats (http://katmat.math. uni-bremen.de/acc/acc.pdf) • Stanford Encyclopedia of Philosophy: " Category Theory (http://plato.stanford.edu/entries/category-theory/)" — by Jean-Pierre Marquis. Extensive bibliography. • List of academic conferences on category theory (http://www.mta.ca/~cat-dist/) • Baez, John, 1996," The Tale of n-categories. (http://math.ucr.edu/home/baez/week73.html)" An informal introduction to higher order categories. • The catsters (http://www.youtube.com/user/TheCatsters) , a Youtube channel about category theory. • Category Theory (http://planetmath.org/?op=getobj&from=objects&id=5622) on PlanetMath • Video archive (http://categorieslogicphysics.wikidot.com/events) of recorded talks relevant to categories, logic and the foundations of physics. • Interactive Web page (http://www.j-paine.org/cgi-bin/webcats/webcats.php) which generates examples of categorical constructions in the category of finite sets. Category (mathematics) In mathematics, a category is an algebraic structure consisting of a collection of "objects", linked together by a collection of "arrows" that have two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Objects and arrows may be abstract entities of any kind. Categories generalize monoids, groupoids and preorders. In addition, the notion of category provides a fundamental and abstract way to describe mathematical entities and their relationships. This is the central idea of category theory, a branch of mathematics which seeks to generalize all of mathematics in terms of objects and arrows, independent of what the objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. For more extensive motivational background and historical notes, see category theory and the list of category theory topics. Two categories are the same if they have the same collection of objects, the same collection of arrows, and the same associative method of composing any pair of arrows. Two categories may also be considered "equivalent" for purposes of category theory, even if they are not precisely the same. Many well-known categories are conventionally identified by a short capitalized word or abbreviation in bold or italics such as Set (category of sets and set [3] functions), Ring (category of rings and ring homomorphisms), or Top (category of topological spaces and continuous maps). Definition A category C consists of • a class ob(C) of objects: • a class hom(C) of morphisms, or arrows, or maps, between the objects. Each morphism/has a unique source object a and target object b where a and b are in ob(C). We write/: a — > b, and we say "/is a morphism from a to b" . We write hom(a, b) (or hom (a, b) when there may be confusion about to which category hom(a, b) refers) to denote the hom-class of all morphisms from a to b. (Some authors write Mor(a, b) or simply C(a, b) instead.) • for every three objects a, b and c, a binary operation hom(a, b) x hom(Z>, c) — > hom(a, c) called composition of morphisms; the composition of/: a — > b and g : b — > c is written as g o/or gf. (Some authors write fg or fig.) such that the following axioms hold: Category (mathematics) 283 • (associativity) iff: a — > b, g : b — > c and h : c — > d then h o (g of) = (h o g) of and • (identity) for every object x, there exists a morphism 1 : x — » x (some authors write id ) called the identity morphism for x, such that for every morphism/: a — > /?, we have 1 of-f-fo 1 . From these axioms, one can prove that there is exactly one identity morphism for every object. Some authors use a slight variation of the definition in which each object is identified with the corresponding identity morphism. History Category theory first appeared in a paper entitled "General Theory of Natural Equivalences", written by Samuel Eilenberg and Saunders Mac Lane in 1945. Small and large categories A category C is called small if both ob(C) and hom(C) are actually sets and not proper classes, and large otherwise. A locally small category is a category such that for all objects a and b, the hom-class hom(a, b) is a set, called a homset. Many important categories in mathematics (such as the category of sets), although not small, are at least locally small. Examples The class of all sets together with all functions between sets, where composition is the usual function composition, forms a large category, Set. It is the most basic and the most commonly used category in mathematics. The category Rel consists of all sets, with binary relations as morphisms. Abstracting from relations instead of functions yields allegories instead of categories. Any class can be viewed as a category whose only morphisms are the identity morphisms. Such categories are called discrete. For any given set 7, the discrete category on I is the small category that has the elements of I as objects and only the identity morphisms as morphisms. Discrete categories are the simplest kind of category. Any preordered set (P, <) forms a small category, where the objects are the members of P, the morphisms are arrows pointing from x to y when x < y. Between any two objects there can be at most one morphism. The existence of identity morphisms and the composability of the morphisms are guaranteed by the reflexivity and the transitivity of the preorder. By the same argument, any partially ordered set and any equivalence relation can be seen as a small category. Any ordinal number can be seen as a category when viewed as a ordered set. Any monoid (any algebraic structure with a single associative binary operation and an identity element) forms a small category with a single object x. (Here, x is any fixed set.) The morphisms from x to x are precisely the elements of the monoid, the identity morphism of x is the identity of the monoid, and the categorical composition of ro] morphisms is given by the monoid operation. Several definitions and theorems about monoids may be generalized for categories. Any group can be seen as a category with a single object in which every morphism is invertible (for every morphism / there is a morphism g that is both left and right inverse to / under composition) by viewing the group as acting on itself by left multiplication. A morphism which is invertible in this sense is called an isomorphism. A groupoid is a category in which every morphism is an isomorphism. Groupoids are generalizations of groups, group actions and equivalence relations. Category (mathematics) 284 Any directed graph generates a small category: the objects are the vertices of the graph, and the morphisms are the paths in the graph (augmented with loops as needed) where composition of morphisms is concatenation of paths. Such a category is called the free category generated by the graph. The class of all preordered sets with monotonic functions as morphisms forms a category, Ord. It is a concrete category, i.e. a category obtained by adding some type of structure onto Set, and requiring that morphisms are functions that respect this added structure. The class of all groups with group homomorphisms as morphisms and function composition as the composition operation forms a large category, Grp. Like Ord, Grp is a concrete category. The category Ab, consisting of all abelian groups and their group homomorphisms, is a full subcategory of Grp, and ri2i the prototype of an abelian category. Other examples of concrete categories are given by the following table. A directed graph. Category Objects Morphisms Mag magmas magma homomorphisms Mar/ smooth manifolds p-times continuously differentiable maps Met metric spaces short maps R-Mod R- Modules, where R is a Ring module homomorphisms Ring rings ring homomorphisms Set sets functions Top topological spaces continuous functions Uni uniform spaces uniformly continuous functions Vect x vector spaces over the field K K- linear maps Fiber bundles with bundle maps between them form a concrete category. The category Cat consists of all small categories, with functors between them as morphisms. Construction of new categories Dual category Any category C can itself be considered as a new category in a different way: the objects are the same as those in the original category but the arrows are those of the original category reversed. This is called the dual or opposite category and is denoted C op . Product categories If C and D are categories, one can form the product category C x D: the objects are pairs consisting of one object from C and one from D, and the morphisms are also pairs, consisting of one morphism in C and one in D. Such pairs can be composed componentwise. Category (mathematics) 285 Types of morphisms A morphism/: a — > b is called a monomorphism (or monic) if fg =fg implies g = g for all morphisms g , g : x — > a. an epimorphism (or epic) if g /= gj implies g = g for all morphisms g , g : b — > x. a bimorphism if it is both a monomorphism and an epimorphism. a retraction if it has a right inverse, i.e. if there exists a morphism g : b — > a with fg = 1 . a section if it has a left inverse, i.e. if there exists a morphism g : b — > a with gf = 1 . a ri3i an isomorphism if it has an inverse, i.e. if there exists a morphism g : b — > a with/g =1 and gf= 1 . an endomorphism if a = b. The class of endomorphisms of a is denoted end(a). an automorphism iff is both an endomorphism and an isomorphism. The class of automorphisms of a is denoted aut(a). Every retraction is an epimorphism. Every section is a monomorphism. The following three statements are equivalent: • /is a monomorphism and a retraction; • /is an epimorphism and a section; • /is an isomorphism. Relations among morphisms (such as fg = h) can most conveniently be represented with commutative diagrams, ri4i where the objects are represented as points and the morphisms as arrows. Types of categories • In many categories, e.g. Ab or Vect , the hom-sets hom(a, b) are not just sets but actually abelian groups, and the composition of morphisms is compatible with these group structures; i.e. is bilinear. Such a category is called preadditive. If, furthermore, the category has all finite products and coproducts, it is called an additive category. If all morphisms have a kernel and a cokernel, and all epimorphisms are cokernels and all monomorphisms are kernels, then we speak of an abelian category. A typical example of an abelian category is the category of abelian groups. • A category is called complete if all limits exist in it. The categories of sets, abelian groups and topological spaces are complete. • A category is called cartesian closed if it has finite direct products and a morphism defined on a finite product can always be represented by a morphism defined on just one of the factors. Examples include Set and CPO, the category of complete partial orders with Scott-continuous functions. • A topos is a certain type of cartesian closed category in which all of mathematics can be formulated (just like classically all of mathematics is formulated in the category of sets). A topos can also be used to represent a logical theory. Category (mathematics) 286 Notes [I] Jacobson (2009), p. 11, ex. 1. [2] Jacobson (2009), p. 12, ex. 9. [3] Jacobson (2009), p. 13, ex. 13. [4] Sica (2006), p. 223; Awodey (2006), p. 1. [5] Jacobson (2009), p. 11, ex. 1. [6] Jacobson (2009), p. 12, ex. 8. [7] Jacobson (2009), p. 13, ex. 12. [8] Jacobson (2009), p. 12, ex. 5. [9] Jacobson (2009), p. 12, ex. 6. [10] Jacobson (2009), p. 12, ex. 7. [II] Jacobson (2009), p. 11, ex. 3. [12] Jacobson (2009), p. 11, ex. 4. [13] Jacobson (2009), p. 12. [14] Jacobson (2009), p. 10. References • Adamek, Jiff; Herrlich, Horst; Strecker, George E. (1990), Abstract and Concrete Categories (http://katmat. math.uni-bremen.de/acc/acc.pdf), John Wiley & Sons, ISBN 0-471-60922-6 (now free on-line edition, GNU FDL). • Asperti, Andrea; Longo, Giuseppe (1991), Categories, Types and Structures (ftp://ftp.di.ens.fr/pub/users/ longo/CategTypesStructures/book.pdf), MIT Press, ISBN 0262011255. • Awodey, Steve (2006), Category theory, Oxford logic guides, 49, Oxford University Press, ISBN 9780198568612. • Barr, Michael; Wells, Charles (2002), Toposes, Triples and Theories (http://www.cwru.edu/artsci/math/wells/ pub/ttt.html), ISBN 0387961151 (revised and corrected free online version of Grundlehren der mathematischen Wissenschaften (278) Springer- Verlag, 1983). • Borceux, Francis (1994), "Handbook of Categorical Algebra", Encyclopedia of Mathematics and its Applications, 50-52, Cambridge: Cambridge University Press, ISBN 0521061199. • Herrlich, Horst; Strecker, George E. (1973), Category Theory, Allen and Bacon, Inc. Boston. • Jacobson, Nathan (2009), Basic algebra (2nd ed.), Dover, ISBN 978-0-486-47187-7. • Lawvere, William; Schanuel, Steve (1997), Conceptual Mathematics: A First Introduction to Categories, Cambridge: Cambridge University Press, ISBN 0521472490. • Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics 5 (2nd ed.), Springer- Verlag, ISBN 0-387-98403-8. • Marquis, Jean-Pierre (2006), "Category Theory" (http://plato.stanford.edu/entries/category-theory/), in Zalta, Edward N., Stanford Encyclopedia of Philosophy (http://plato.stanford.edu). • Sica, Giandomenico (2006), What is category theory?, Advanced studies in mathematics and logic, 3, Polimetrica, ISBN 9788876990311. Category (mathematics) 287 External links • Chris Hillman, Categorical primer (http://citeseer.ist.psu.edU/cache/papers/cs/23543/http:zSzzSzwww-aix. gsi.dezSz~appelzSzskriptezSzotherzSzcategories.pdf/hillman01categorical.pdf), formal introduction to Category Theory. • Homepage of the Categories mailing list (http://www.mta.ca/~cat-dist/categories.html), with extensive list of resources • Category Theory section of Alexandre Stefanov's list of free online mathematics resources (http://web. archive. org/web/20091027042032/http://us. geocities.com/alex_stef/mylist.html) Glossary of category theory This is a glossary of properties and concepts in category theory in mathematics. Categories A category A is said to be: • small provided that the class of all morphisms is a set (i.e., not a proper class); otherwise large. • locally small provided that the morphisms between every pair of objects A and B form a set. • Some authors assume a foundation in which the collection of all classes forms a "conglomerate", in which case a quasicategory is a category whose objects and morphisms merely form a conglomerate. (NB other authors use the term "quasicategory" with a different meaning. ) isomorphic to a category B provided that there is an isomorphism between them. equivalent to a category B provided that there is an equivalence between them. concrete provided that there is a faithful functor from A to Set; e.g., Vec, Grp and Top. discrete provided that each morphism is an identity morphism (of some object). thin category provided that there is at most one morphism between any pair of objects, a subcategory of a category B provided that there is an inclusion functor given from A to B. a full subcategory of a category B provided that the inclusion functor is full. wellpowered provided for each object A there is only a set of pairwise non-isomorphic subobjects. complete provided that all small limits exist. cartesian closed provided that it has a terminal object and that any two objects have a product and exponential. abelian provided that it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal. r3i normal provided that every monic is normal. balanced if every bimorphism is an isomorphism. R -linear (R is a commutative ring) if A is locally small, each hom set is an 7?-module, and composition of morphisms is 7?-bilinear. The category A is also said to be over R. Glossary of category theory 288 Morphisms A morphism/in a category is called: • an epimorphism provided that g = h whenever gof = hof.In other words, /is the dual of a monomorphism. • an identity provided that/maps an object A to A and for any morphisms g with domain A and h with codomain A, 9 ° f = 9 and f o h = h. • an inverse to a morphism g if g o /is defined and is equal to the identity morphism on the domain off, and f o gis defined and equal to the identity morphism on the codomain of g. The inverse of g is unique and is denoted by g ~ • an isomorphism provided that there exists an inverse off. • a monomorphism (also called monk) provided that g = h whenever fog = f®h; e.g., an injection in Set. In other words, /is the dual of an epimorphism. Functors A functor F is said to be: • a constant provided that F maps every object in a category to the same object A and every morphism to the identity on A. • faithful provided that F is injective when restricted to each hom-set. • full provided that F is surjective when restricted to each hom-set. • isomorphism-dense (sometimes called essentially surjective) provided that for every B there exists A such that F(A) is isomorphic to B. • an equivalence provided that F is faithful, full and isomorphism-dense. • amnestic provided that if k is an isomorphism and F{k) is an identity, then k is an identity. • reflect identities provided that if F(k) is an identity then k is an identity as well. • reflect isomorphisms provided that if F(k) is an isomorphism then k is an isomorphism as well. Objects An object A in a category is said to be: • isomorphic to an object B provided that there is an isomorphism between A and B. • initial provided that there is exactly one morphism from A to each object B; e.g., empty set in Set. • terminal provided that there is exactly one morphism from each object B to A; e.g., singletons in Set. • a zero object if it is both initial and terminal, such as a trivial group in Grp. An object A in an abelian category is: • simple if it is not isomorphic to the zero object and any subobject of A is isomorphic to zero or to A. • finite length if it has a composition series. The number of proper subobjects in any such composition series is called the length of A. Glossary of category theory 289 Notes [1] Adamek, Jiff; Herrlich, Horst, and Strecker, George E (2004) [1990] (PDF). Abstract and Concrete Categories (The Joy of Cats) (http:// katmat.math.uni-bremen.de/acc/). New York: Wiley & Sons. p. 40. ISBN 0-471-60922-6. . [2] Joyal, A. (2002). "Quasi-categories and Kan complexes". Journal of Pure and Applied Algebra 175: 207—222. doi:10.1016/S0022-4049(02)00135-4. [3] http://planetmath.org/encyclopedia/NormalCategory.html [4] Kashiwara & Schapira 2006, exercise 8.20 References • Kashiwara, Masaki; Schapira, Pierre (2006), Categories and sheaves Dual (category theory) In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category C op . Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category C op . Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement is true about C, then its dual statement is true about C op . Also, if a statement is false about C, then its dual has to be false about C op . Given a concrete category C, it is often the case that the opposite category C op per se is abstract. C° p need not be a category that arises from mathematical practice. In this case, another category D is also termed to be in duality with C if D and C op are equivalent as categories. In the case when C and its opposite C op are equivalent, such a category is self-dual. Formal definition We define the elementary language of category theory as the two-sorted first order language with objects and morphisms as distinct sorts, together with the relations of an object being the source or target of a morphism and a symbol for composing two morphisms. Let o be any statement in this language. We form the dual o op as follows: 1. Interchange each occurrence of "source" in o with "target". 2. Interchange the order of composing morphisms. That is, replace each occurrence of g o /with fog Informally, these conditions state that the dual of a statement is formed by reversing arrows and compositions. Duality is the observation that o is true for some category C if and only if o op is true for C op . Examples • A morphism f: A — > B is a monomorphism iffog = foh implies g = h . Performing the dual operation, we get the statement that go/ = fto/ implies g = h.for a morphism f'.B — > A . This is precisely what it means for/to be an epimorphism. In short, the property of being a monomorphism is dual to the property of being an epimorphism. Applying duality, this means that a morphism in some category C is a monomorphism if and only if the reverse morphism in the opposite category C op is an epimorphism. • An example comes from reversing the direction of inequalities in a partial order. So if X is a set and < a partial order relation, we can define a new partial order relation < by new x < y if and only if y < x. new J Dual (category theory) 290 This example on orders is a special case, since partial orders correspond to a certain kind of category in which Hom(A,B) can have at most one element. In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). For example, if we take the opposite of a lattice, we will find that meets and joins have their roles interchanged. This is an abstract form of De Morgan's laws, or of duality applied to lattices. • Limits and colimits are dual notions. • Fibrations and cofibrations are examples of dual notions in algebraic topology and homotopy theory. In this context, the duality is often called Eckmann— Hilton duality. Abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative attempt to unify several cohomology theories by Alexander Grothendieck. Abelian categories are very stable categories, for example they are regular and they satisfy the snake lemma. The class of Abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an Abelian category, or the category of functors from a small category to an Abelian category are Abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Definitions A category is abelian if • it has a zero object, • it has all pullbacks and pushouts, and • all monomorphisms and epimorphisms are normal. By a theorem of Peter Freyd, this definition is equivalent to the following "piecemeal" definition: • A category is preadditive if it is enriched over the monoidal category Ab of abelian groups. This means that all hom-sets are abelian groups and the composition of morphisms is bilinear. • A preadditive category is additive if every finite set of objects has a biproduct. This means that we can form finite direct sums and direct products. • An additive category is preabelian if every morphism has both a kernel and a cokernel. • Finally, a preabelian category is abelian if every monomorphism and every epimorphism is normal. This means that every monomorphism is a kernel of some morphism, and every epimorphism is a cokernel of some morphism. Note that the enriched structure on hom-sets is a consequence of the three axioms of the first definition. This highlights the foundational relevance of the category of Abelian groups in the theory and its canonical nature. The concept of exact sequence arises naturally in this setting, and it turns out that exact functors, i.e. the functors preserving exact sequences in various senses, are the relevant functors between Abelian categories. This exactness concept has been axiomatized in the theory of exact categories, forming a very special case of regular categories. Abelian category 291 Examples • As mentioned above, the category of all abelian groups is an abelian category. The category of all finitely generated abelian groups is also an abelian category, as is the category of all finite abelian groups. • If R is a ring, then the category of all left (or right) modules over R is an abelian category. In fact, it can be shown that any small abelian category is equivalent to a full subcategory of such a category of modules (Mitchell's embedding theorem). • If R is a left-noetherian ring, then the category of finitely generated left modules over R is abelian. In particular, the category of finitely generated modules over a noetherian commutative ring is abelian; in this way, abelian categories show up in commutative algebra. • As special cases of the two previous examples: the category of vector spaces over a fixed field k is abelian, as is the category of finite-dimensional vector spaces over lc. • If X is a topological space, then the category of all (real or complex) vector bundles on X is not usually an abelian category, as there can be monomorphisms that are not kernels. • If X is a topological space, then the category of all sheaves of abelian groups on X is an abelian category. More generally, the category of sheaves of abelian groups on a Grothendieck site is an abelian category. In this way, abelian categories show up in algebraic topology and algebraic geometry. • If C is a small category and A is an abelian category, then the category of all functors from C to A forms an abelian category (the morphisms of this category are the natural transformations between functors). If C is small and preadditive, then the category of all additive functors from C to A also forms an abelian category. The latter is a generalization of the 7?-module example, since a ring can be understood as a preadditive category with a single object. Grothendieck's axioms In his Tohoku article, Grothendieck listed four additional axioms (and their duals) that an abelian category A might satisfy. These axioms are still in common use to this day. They are the following: • AB3) For every set {A.} of objects of A, the coproduct |JA. exists in A (i.e. A is cocomplete). • AB4) A satisfies AB3), and the coproduct of a family of monomorphisms is a monomorphism. • AB5) A satisfies AB3), and filtered colimits of exact sequences are exact. and their duals • AB3*) For every set {A.} of objects of A, the product I1A. exists in A (i.e. A is complete). • AB4*) A satisfies AB3*), and the product of a family of epimorphisms is an epimorphism. • AB5*) A satisfies AB3*), and filtered limits of exact sequences are exact. Axioms AB1) and AB2) were also given. They are what make an additive category abelian. Specifically: • AB 1) Every morphism has a kernel and a cokernel. • AB2) For every morphism/, the canonical morphism from coim/to im/is an isomorphism. Grothendieck also gave axioms AB6) and AB6*). Abelian category 292 Elementary properties Given any pair A, B of objects in an abelian category, there is a special zero morphism from A to B. This can be defined as the zero element of the hom-set Hom(A,B), since this is an abelian group. Alternatively, it can be defined as the unique composition A — > — > B, where is the zero object of the abelian category. In an abelian category, every morphism / can be written as the composition of an epimorphism followed by a monomorphism. This epimorphism is called the coimage off, while the monomorphism is called the image off. Subobjects and quotient objects are well-behaved in abelian categories. For example, the poset of subobjects of any given object A is a bounded lattice. Every abelian category A is a module over the monoidal category of finitely generated abelian groups; that is, we can form a tensor product of a finitely generated abelian group G and any object A of A. The abelian category is also a comodule; Hom(GA) can be interpreted as an object of A. If A is complete, then we can remove the requirement that G be finitely generated; most generally, we can form finitary enriched limits in A. Related concepts Abelian categories are the most general setting for homological algebra. All of the constructions used in that field are relevant, such as exact sequences, and especially short exact sequences, and derived functors. Important theorems that apply in all abelian categories include the five lemma (and the short five lemma as a special case), as well as the snake lemma (and the nine lemma as a special case). History Abelian categories were introduced by Buchsbaum (1955) (under the name of "exact category") and Grothendieck (1957) in order to unify various cohomology theories. At the time, there was a cohomology theory for sheaves, and a cohomology theory for groups. The two were defined differently, but they had similar properties. In fact, much of category theory was developed as a language to study these similarities. Grothendieck unified the two theories: they both arise as derived functors on abelian categories; the abelian category of sheaves of abelian groups on a topological space, and the abelian category of G-modules for a given group G. References • Buchsbaum, D. A. (1955), "Exact categories and duality" , Transactions of the American Mathematical Society 80 (1): 1-34, MR0074407, ISSN 0002-9947 121 • Freyd, Peter (1964), Abelian Categories , New York: Harper and Row T31 • Grothendieck, Alexander (1957), "Sur quelques points d'algebre homologique" , The Tohoku Mathematical Journal. Second Series 9: 119-221, MR0102537, ISSN 0040-8735 • Mitchell, Barry (1965), Theory of Categories, Boston, MA: Academic Press • Popescu, N. (1973), Abelian categories with applications to rings and modules, Boston, MA: Academic Press References [1] http://www.jstor.org/stable/1993003 [2] http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html [3] http://projecteuclid.org/euclid.tmj/! 178244839 Yoneda lemma 293 Yoneda lemma In mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cay ley's theorem from group theory (viewing a group as a particular kind of category with just one object). It allows the embedding of any category into a category of functors defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda. Generalities The Yoneda lemma suggests that instead of studying the (small) category C, one should study the category of all functors of C into Set (the category of sets with functions as morphisms). Set is a category we understand well, and a functor of C into Set can be seen as a "representation" of C in terms of known structures. The original category C is contained in this functor category, but new objects appear in the functor category which were absent and "hidden" in C. Treating these new objects just like the old ones often unifies and simplifies the theory. This approach is akin to (and in fact generalizes) the common method of studying a ring by investigating the modules over that ring. The ring takes the place of the category C, and the category of modules over the ring is a category of functors defined on C. Formal statement General version Yoneda's lemma concerns functors from a fixed category C to the category of sets, Set. If C is a locally small category (i.e. the hom-sets are actual sets and not proper classes), then each object A of C gives rise to a natural functor to Set called a hom-functor. This functor is denoted: h A = Hom(A -), The hom-functor h sends X to the set of morphisms Hom(A,X) and sends a morphism/ from X to Y to the morphism fo — (composition with/on the left) that sends a morphism g in Hom(A,X) to the morphism/o g in Hom(A,Y). Let F be an arbitrary functor from C to Set. Then Yoneda's lemma says that for each object A of C, the natural transformations from h to F are in one-to-one correspondence with the elements of F{A). That is, Na.t(h A ,F)^F(A). Given a natural transformation O from h to F, the corresponding element of F(A) is u = t&^fidyi) • There is a contravariant version of Yoneda's lemma which concerns contravariant functors from C to Set. This version involves the contravariant hom-functor fiA = Hom(— , A), which sends X to the hom-set Hom(X,A). Given an arbitrary contravariant functor G from C to Set, Yoneda's lemma asserts that Na,t(h A ,G) = G(A). Yoneda lemma 294 Naming conventions The use of "h " for the covariant hom-functor and "h " for the contravariant hom-functor is not completely standard. However, the exceptions to this rule are usually completely unrelated symbols, and it is very rare to see "h " used to mean the covariant hom-functor or vice-versa. The mnemonic "falling into something" can be helpful in remembering that "h " is the contravariant hom-functor. When the letter "A" is falling (i.e. a subscript), h assigns to an object X the morphisms from X into A. Proof The proof of Yoneda's lemma is indicated by the following commutative diagram: Hom(AX) H ° lll(A/) - llom(AA) * icU .. h (Fj>=*je(/) ®: F(A) Ff F(X) This diagram shows that the natural transformation O is completely determined by ^^fid^) = U since for each morphism/: A — > X one has Moreover, any element u€F(A) defines a natural transformation in this way. The proof in the contravariant case is completely analogous. In this way, Yoneda's Lemma provides a complete classification of all natural transformations from the functor Hom(A,-) to an arbitrary functor F:C— »Set. The Yoneda embedding ,B An important special case of Yoneda's lemma is when the functor F from C to Set is another hom-functor h . In this case, the covariant version of Yoneda's lemma states that Nat(/i A ,h s ) = Hom(S,A). That is, natural transformations between hom-functors are in one-to-one correspondence with morphisms (in the reverse direction) between the associated objects. Given a morphism/: B — > A the associated natural transformation is denoted Hom^,— ). Mapping each object A in C to its associated hom-functor h = Hom(A,— ) and each morphism / : B — > A to the corresponding natural transformation Hom(f,— ) determines a contravariant functor h from C to Set , the functor category of all (covariant) functors from C to Set. One can interpret h as a covariant functor: br-.c op Set L Yoneda lemma 295 The meaning of Yoneda's lemma in this setting is that the functor h is fully faithful, and therefore gives an embedding of C op in the category of functors to Set. The collection of all functors {h , A in C} is a subcategory of Set . Therefore, Yoneda embedding implies that the category C op is isomorphic to the category {h , A in C}. The contravariant version of Yoneda's lemma states that Nat(h A ,h B ) = Hom(,4,S). Therefore, h gives rise to a covariant functor from C to the category of contravariant functors to Set: h_:C ^Set c ° P . Yoneda's lemma then states that any locally small category C can be embedded in the category of contravariant functors from C to Set via h_. This is called the Yoneda embedding. Preadditive categories, rings and modules A preadditive category is a category where the morphism sets form abelian groups and the composition of morphisms is bilinear; examples are categories of abelian groups or modules. In a preadditive category, there is both a "multiplication" and an "addition" of morphisms, which is why preadditive categories are viewed as generalizations of rings. Rings are preadditive categories with one object. The Yoneda lemma remains true for preadditive categories if we choose as our extension the category of additive contravariant functors from the original category into the category of abelian groups; these are functors which are compatible with the addition of morphisms and should be thought of as forming a module category over the original category. The Yoneda lemma then yields the natural procedure to enlarge a preadditive category so that the enlarged version remains preadditive — in fact, the enlarged version is an abelian category, a much more powerful condition. In the case of a ring R, the extended category is the category of all left modules over R, and the statement of the Yoneda lemma reduces to the well-known isomorphism M = Uom(R,M) for all left modules M over R. R Notes [1] A notable exception is Commutative algebra with a view toward algebraic geometry' I David Eisenbud (1995), which uses "h " to mean the covariant hom-functor. However, the later book The geometry of schemes I David Eisenbud, Joe Harris (1998) reverses this and uses "h " to mean the contravariant hom-functor. References • Freyd, Peter (1964), Abelian categories, Harper and Row. Reprinted 2003 (http://www.tac.mta.ca/tac/ reprints/articles/3/tr3abs.html). • Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics, 5, Springer- Verlag, ISBN 0-387-98403-8. Limit (category theory) 296 Limit (category theory) In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits. Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize. Definition Limits and colimits in a category C are defined by means of diagrams in C. Formally, a diagram of type / in C is a functor from / to C: F : J -» C. The category J is thought of as index category, and the diagram F is thought of as indexing a collection of objects and morphisms in C patterned on J. The actual objects and morphisms in J are largely irrelevant — only the way in which they are interrelated matters. One is most often interested in the case where the category J is a small or even finite category. A diagram is said to be small or finite whenever J is. Limits Let F : J — > C be a diagram of type J in a category C. A cone to F is an object N of C together with a family of morphisms one for each object X of /, such that for every morphism/: X — > Yin J, we have F(f) o op = ty v A limit of the diagram F : J — > C is a cone (L, cp) to F such that for any other cone (N, i|>) to F there exists a unique morphism u : N — > L such that cp o u - i|) f or all X in J. Limit (category theory) 297 One says that the cone (N, op) factors through the cone (L, cp) with the unique factorization u. The morphism u is sometimes called the mediating morphism. Limits are also referred to as universal cones, since they are characterized by a universal property (see below for more information). As with every universal property, the above definition describes a balanced state of generality: The limit object L has to be general enough to allow any other cone to factor through it; on the other hand, L has to be sufficiently specific, so that only one such factorization is possible for every cone. Limits may also be characterized as terminal objects in the category of cones to F. It is possible that a diagram does not have a limit at all. However, if a diagram does have a limit then this limit is essentially unique: it is unique up to a unique isomorphism. For this reason one often speaks of the limit of F. Colimits The dual notions of limits and cones are colimits and co-cones. Although it is straightforward to obtain the definitions of these by inverting all morphisms in the above definitions, we will explicitly state them here: A co-cone of a diagram F : J — > C is an object N of C together with a family of morphisms y x :F(X)->N for every object X of J, such that for every morphism/: X — » Yin J, we have i|> o F(f)= i|) . A colimit of a diagram F : J — > C is a co-cone (L, (j) ) of F such that for any other co-cone (N, op) of F there exists a unique morphism u : L — > N such that mo (j) - ajj for all X in J. Limit (category theory) 298 Colimits are also referred to as universal co-cones. They can be characterized as initial objects in the category of co-cones from F. As with limits, if a diagram F has a colimit then this colimit is unique up to a unique isomorphism. Variations Limits and colimits can also be defined for collections of objects and morphisms without the use of diagrams. The definitions are the same (note that in definitions above we never needed to use composition of morphisms in J). This variation, however, adds no new information. Any collection of objects and morphisms defines a (possibly large) directed graph G. If we let / be the free category generated by G, there is a universal diagram F : J — > C whose image contains G. The limit (or colimit) of this diagram is the same as the limit (or colimit) of the original collection of objects and morphisms. Weak limit and weak colimits are defined like limits and colimits, except that the uniqueness property of the mediating morphism is dropped. Limit (category theory) 299 Examples Limits The definition of limits is general enough to subsume several constructions useful in practical settings. In the following we will consider the limit (L, cp) of a diagram F : J — > C. • Terminal objects. If J is the empty category there is only one diagram of type /: the empty one (similar to the empty function in set theory). A cone to the empty diagram is essentially just an object of C. The limit of F is any object that has a unique factorization through any other object. This is just the definition of a terminal object. • Products. If J is a discrete category then a diagram F is essentially nothing but a family of objects of C, indexed by /. The limit L of F is called the product of these objects. The cone cp consists of a family of morphisms cp : L — » F(X) called the projections of the product. In the category of sets, for instance, the products are given by Cartesian products and the projections are just the natural projections onto the various factors. • Powers. A special case of a product is when the diagram F is a constant functor to an object X of C. The limit th J of this diagram is called the / power oiX and denoted X . • Equalizers. If / is a category with two objects and two parallel morphisms from object 1 to object 2 then a diagram of type J is a pair of parallel morphisms in C. The limit L of such a diagram is called an equalizer of those morphisms. • Kernels. A kernel is a special case of an equalizer where one of the morphisms is a zero morphism. • Fullbacks. Let F be a diagram that picks out three objects X, Y, and Z in C, where the only non-identity morphisms are/: X — > Z and g : Y — > Z. The limit L of F is called apullback or a fiber product. It can nicely be visualized as a commutative square: Inverse limits. Let J be a directed poset (considered as a small category by adding arrows i — > j if and only if i < j) and let F : 7° p — > C be a diagram. The limit of F is called (confusingly) an inverse limit, projective limit, or directed limit. If J = 1, the category with a single object and morphism, then a diagram of type / is essentially just an object of C. A cone to an object X is just a morphism with codomain X. A morphism/ : Y — > X is a limit of the diagram X if and only if/ is an isomorphism. More generally, if /is any category with an initial object i, then any diagram of type / has a limit, namely any object isomorphic to F(i). Such an isomorphism uniquely determines a universal cone to F. Limit (category theory) 300 Colimits Examples of colimits are given by the dual versions of the examples above: • Initial objects are colimits of empty diagrams. • Coproducts are colimits of diagrams indexed by discrete categories. • Copowers are colimits of constant diagrams from discrete categories. • Coequalizers are colimits of a parallel pair of morphisms. • Cokernels are coequalizers of a morphism and a parallel zero morphism. • Pushouts are colimits of a pair of morphisms with common domain. • Direct limits are colimits of diagrams indexed by directed sets. Properties Existence of limits A given diagram F : J — > C may or may not have a limit (or colimit) in C. Indeed, there may not even be a cone to F, let alone a universal cone. A category C is said to have limits of type J if every diagram of type J has a limit in C. Specifically, a category C is said to • have products if it has limits of type / for every small discrete category J (it need not have large products), • have equalizers if it has limits of type • n£ • (i.e. every parallel pair of morphisms has an equalizer), • have pullbacks if it has limits of type • — > • < — • (i.e. every pair of morphisms with common codomain has a pullback). A complete category is a category that has all small limits (i.e. all limits of type / for every small category J). One can also make the dual definitions. A category has colimits of type J if every diagram of type / has a colimit in C. A cocomplete category is one that has all small colimits. The existence theorem for limits states that if a category C has equalizers and all products indexed by the classes Ob(/) and Hom(/), then C has all limits of type J. In this case, the limit of a diagram F : J — > C can be constructed as the equalizer of the two morphisms s,t: ft F i= * J] F cod(/) ieOb(J) /eHom(J) given (in component form) by S= (F(f) ° 7Tdom(/)} * = (^codCf)}- There is a dual existence theorem for colimits in terms of coequalizers and coproducts. Both of these theorems give sufficient but not necessary conditions for the existence of all (co)limits of type J. Limit (category theory) 301 Universal property Limits and colimits are important special cases of universal constructions. Let C be a category and let J be a small index category. The functor category (J may be thought of the category of all diagrams of type J in C. The diagonal functor is the functor that maps each object N in C to the constant functor A(A0 : J — > C to N. That is, A(N)(X) = N for each object X in / and A(N)(f) = id for each morphism/in J. Given a diagram F: J —> C (thought of as an object in Cr), a natural transformation i|> : A(A0 — > F (which is just a morphism in the category (s) is the same thing as a cone from N to F. The components of op are the morphisms i|> : iV — > F (X). Dually, a natural transformation op : F — > A(A0 is the same thing as a co-cone from F to iV. The definitions of limits and colimits can then be restated in the form: • A limit of F is a universal morphism from A to F. • A colimit of F is a universal morphism from F to A. Adjunctions Like all universal constructions, the formation of limits and colimits is functorial in nature. In other words, if every diagram of type / has a limit in C (for / small) there exists a limit functor lim :C J -> C which assigns each diagram its limit and each natural transformation r) : F — > G the unique morphism lim 11 : lim F — > lim G commuting with the corresponding universal cones. This functor is right adjoint to the diagonal functor A : C —> C . This adjunction gives a bijection between the set of all morphisms from N to lim F and the set of all cones from N to F Hom(iV, limF) - Cone(iV, F) which is natural in the variables N and F. The counit of this adjunction is simply the universal cone from lim F to F. If the index category J is connected (and nonempty) then the unit of the adjunction is an isomorphism so that lim is a left inverse of A. This fails if / is not connected. For example, if J is a discrete category, the components of the unit are the diagonal morphisms 5 : N —> N . Dually, if every diagram of type / has a colimit in C (for / small) there exists a colimit functor colim : C —> C which assigns each diagram its colimit. This functor is left adjoint to the diagonal functor A : C —> (J , and one has a natural isomorphism Hom(colimF, N) ^ Cocone(F, N). The unit of this adjunction is the universal cocone from F to colim F. If J is connected (and nonempty) then the counit is an isomorphism, so that colim is a left inverse of A. Note that both the limit and the colimit functors are covariant functors. Limit (category theory) 302 As representations of functors One can use Horn functors to relate limits and colimits in a category C to limits in Set, the category of sets. This follows, in part, from the fact the covariant Horn functor Hom(A', — ) : C — > Set preserves all limits in C. By duality, the contravariant Horn functor must take colimits to limits. If a diagram F : J — > C has a limit in C, denoted by lim F, there is a canonical isomorphism Hom(iV, limF) = lim Hom(iV, F-) which is natural in the variable N. Here the functor Hom(iV, F-) is the composition of the Horn functor Hom(iV, — ) with F. This isomorphism is the unique one which respects the limiting cones. One can use the above relationship to define the limit of F in C. The first step is to observe that the limit of the functor Hom(iV, F-) can be identified with the set of all cones from N to F: limHom(iV, F-) = Cone(i\T, F). The limiting cone is given by the family of maps n : Cone(N, F) — » Hom(N, FX) where jt (i|0 = i|> . If one is given an object L of C together with a natural isomorphism O : Hom(— , L) — > Cone(— , F), the object L will be a limit of F with the limiting cone given by O (id ). In fancy language, this amounts to saying that a limit of F is a representation of the functor Cone(— , F) : C — » Set. Dually, if a diagram F : J — > C has a colimit in C, denoted colim F, there is a unique canonical isomorphism Hom(colimF, TV) = limHom(F-, N) which is natural in the variable N and respects the colimiting cones. Identifying the limit of Hom(F— , N) with the set Cocone(F, N), this relationship can be used to define the colimit of the diagram F as a representation of the functor Cocone(F, — ). Interchange of limits and colimits of sets Let / be a finite category and J be a small filtered category. For any bifunctor F : I x J -> Set there is a natural isomorphism colim j limjF(i, j) — » lim/ colimj F{i, j) . In words, filtered colimits in Set commute with finite limits. Functors and limits If F : J — > C is a diagram in C and G : C — > D is a functor then by composition (recall that a diagram is just a functor) one obtains a diagram GF : J — > D. A natural question is then: "How are the limits of GF related to those of FT Preservation of limits A functor G : C — > D induces a map from Cone(F) to Cone(GF): if W is a cone from N to F then G'P is a cone from GN to GF. The functor G is said to preserve the limits of F if (GL, Gcp) is a limit of GF whenever (L, cp) is a limit of F. (Note that if the limit of F does not exist, then G vacuously preserves the limits of F.) A functor G is said to preserve all limits of type J if it preserves the limits of all diagrams F : J — > C. For example, one can say that G preserves products, equalizers, pullbacks, etc. A continuous functor is one that preserves all small limits. One can make analogous definitions for colimits. For instance, a functor G preserves the colimits of F if G(L, cp) is a colimit of GF whenever (L, cp) is a colimit of F. A cocontinuous functor is one that preserves all small colimits. Limit (category theory) 303 If C is a complete category, then, by the above existence theorem for limits, a functor G : C — > D is continuous if and only if it preserves (small) products and equalizers. Dually, G is cocontinuous if and only if it preserves (small) coproducts and coequalizers. An important property of adjoint functors is that every right adjoint functor is continuous and every left adjoint functor is cocontinuous. Since adjoint functors exist in abundance, this gives numerous examples of continuous and cocontinuous functors. For a given diagram F : J — > C and functor G : C — > D, if both F and GF have specified limits there is a unique canonical morphism t r : G lim F -> lim GF F which respects the corresponding limit cones. The functor G preserves the limits of F if and only this map is an isomorphism. If the categories C and D have all limits of type / then lim is a functor and the morphisms x form the F components of a natural transformation t : G lim — > lim G . The functor G preserves all limits of type / if and only if t is a natural isomorphism. In this sense, the functor G can be said to commute with limits (up to a canonical natural isomorphism). Preservation of limits and colimits is a concept that only applies to covariant functors. For contravariant functors the corresponding notions would be a functor that takes colimits to limits, or one that takes limits to colimits. Lifting of limits A functor G : C — > D is said to lift limits for a diagram F : J — » C if whenever (L, cp) is a limit of GF there exists a limit (L', cp') of F such that G(L', cp') = (L, cp). A functor G lifts limits of type J if it lifts limits for all diagrams of type /. One can therefore talk about lifting products, equalizers, pullbacks, etc. Finally, one says that G lifts limits if it lifts all limits. There are dual definitions for the lifting of colimits. A functor G lifts limits uniquely for a diagram F if the preimage cone (IS, cp') is the unique limit of F such that G(L', cp') = (L, cp). One can show that G lifts limits uniquely if and only if it lifts limits and is amnestic. Lifting of limits is clearly related to preservation of limits. If G lifts limits for a diagram F and GF has a limit, then F also has a limit and G preserves the limits of F. It follows that: • If G lifts limits of all type J and D has all limits of type /, then C also has all limits of type / and G preserves these limits. • If G lifts all small limits and D is complete, then C is also complete and G is continuous. The dual statements for colimits are equally valid. Creation and reflection of limits Let F : J — > C be a diagram. A functor G : C — > D is said to • create limits for F if whenever (L, cp) is a limit of GF there exists a unique cone (IS, cp') to F such that G(L', cp') = (L, cp), and furthermore, this cone is a limit of F. • reflect limits for F if each cone to F whose image under G is a limit of GF is already a limit of F. Dually, one can define creation and reflection of colimits. The following statements are easily seen to be equivalent: • The functor G creates limits. • The functor G lifts limits uniquely and reflects limits. There are examples of functors which lift limits uniquely but neither create nor reflect them. Limit (category theory) 304 Examples • For any category C and object A of C the Horn functor Hom(A,— ) : C — > Set preserves all limits in C. In particular, Horn functors are continuous. Horn functors need not preserve colimits. • Every representable functor C — > Set preserves limits (but not necessarily colimits). • The forgetful functor U : Grp — » Set creates (and preserves) all small limits and filtered colimits; however, U does not preserve coproducts. This situation is typical of algebraic forgetful functors. • The free functor F : Set — > Grp (which assigns to every set S the free group over S) is left adjoint to forgetful functor U and is, therefore, cocontinuous. This explains why the free product of two free groups G and H is the free group generated by the disjoint union of the generators of G and H. • The inclusion functor Ab — > Grp creates limits but does not preserve coproducts (the coproduct of two abelian groups being the direct sum). • The forgetful functor Top — > Set lifts limits and colimits uniquely but creates neither. • Let Met be the category of metric spaces with continuous functions for morphisms. The forgetful functor Met — » Set lifts finite limits but does not lift them uniquely. A note on terminology Older terminology referred to limits as "inverse limits" or "projective limits," and to colimits as "direct limits" or "inductive limits." This has been the source of a lot of confusion. There are several ways to remember the modern terminology. First of all, • cokernels, • coequalizers, and • codomains are types of colimits, whereas • kernels, • equalizers, and • domains are types of limits. Second, the prefix "co" implies "first variable of the Horn "■ Terms like "cohomology" and "cofibration" all have a slightly stronger association with the first variable, i.e., the contravariant variable, of the Horn bifunctor. References • Adamek, Jiff; Horst Herrlich, and George E. Strecker (1990). Abstract and Concrete Categories . John Wiley & Sons. ISBN 0-471-60922-6. • Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 (2nd ed. ed.). Springer. ISBN 0-387-98403-8. External links • Interactive Web page which generates examples of limits and colimits in the category of finite sets. Written by Jocelyn Paine . Limit (category theory) 305 References [1] http://katmat.math.uni-bremen.de/acc/acc.pdf [2] http://www.j-paine.org/cgi-bin/webcats/webcats.php [3] http://www.j-paine.org/ Adjoint functors In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency. It is studied in generality by the branch of mathematics known as category theory, which helps to minimize the repetition of the same logical details separately in every subject. In the most concise symmetric definition, an adjunction between categories C and D is a pair of functors, F \V ->C and Q : Q -► V and a family of bijections hom c (Fy, X) = liomp(y, GX) which is natural in the variables X and Y. The functor F is called a left adjoint functor, while G is called a right adjoint functor. The relationship "F is left adjoint to G" (or equivalently, "G is right adjoint to F") is sometimes written This definition and others are made precise below. Introduction "The slogan is 'Adjoint functors arise everywhere'." (Saunders Mac Lane, Categories for the working mathematician) The long list of examples in this article is only a partial indication of how often an interesting mathematical construction is an adjoint functor. As a result, general theorems about left/right adjoint functors, such as the equivalence of their various definitions or the fact that they respectively preserve colimits/limits (which are also found in every area of math), can encode the details of many useful and otherwise non-trivial results. Motivation One good way to motivate adjoint functors is to vaguely explain what problem they solve, and how they solve it. (This motivation runs parallel to the definitions via universal morphisms below.) Adjoint functors as formulaic solutions to optimization problems It can be said that an adjoint functor is a way of giving the most efficient solution to some problem via a method which is formulaic. For example, an elementary problem in ring theory is how to turn a rng (which is like a ring that might not have a multiplicative identity) into a ring. The most efficient way is to adjoin an element '1' to the rng, adjoin no unnecessary extra elements (we will need to have r+\ for each r in the ring, clearly), and impose no relations in the newly formed ring that are not forced by axioms. Moreover, this construction is formulaic in the sense that it works in essentially the same way for any rng. This is rather vague, though suggestive, and can be made precise in the language of category theory: a construction is most efficient if it satisfies a universal property, and is formulaic if it defines a functor. Universal properties come in two types: initial properties and terminal properties. Since these are dual (opposite) notions, it is only necessary to discuss one of them. Adjoint functors 306 The idea of using an initial property is to set up the problem in terms of some auxiliary category E, and then identify that what we want is to find an initial object of E. This has an advantage that the optimization — the sense that we are finding the most efficient solution — means something rigorous and is recognisable, rather like the attainment of a supremum. Picking the right category E is something of a knack: for example, take the given rng R, and make a category E whose objects are rng homomorphisms R — > S, with S a ring having a multiplicative identity. The morphisms in E are commutative triangles of the form (R — > S ,R — > S , S — > S ) where S — > S is a ring map (which preserves the identity). The assertion that an object i? — > 7?* is initial in E means that the ring R* is a most efficient solution to our problem. The two facts that this method of turning rngs into rings is most efficient and formulaic can be expressed simultaneously by saying that it defines an adjoint functor. The hidden symmetry of optimization problems Continuing this discussion, suppose we started with the functor F, and posed the following (vague) question: is there a problem to which F is the most efficient solution? The notion that F is the most efficient solution to the problem posed by G is, in a certain rigorous sense, equivalent to the notion that G poses the most difficult problem which F solves. This has the intuitive meaning that adjoint functors should occur in pairs, and in fact they do, but this is not trivial from the universal morphism definitions. The equivalent symmetric definitions involving adjunctions and the symmetric language of adjoint functors (we can say either F is left adjoint to G or G is right adjoint to F) have the advantage of making this fact explicit. Formal definitions There are various definitions for adjoint functors. Their equivalence is elementary but not at all trivial and in fact highly useful. This article provides several such definitions: • The definitions via universal morphisms are easy to state, and require minimal verifications when constructing an adjoint functor or proving two functors are adjoint. They are also the most analogous to our intuition involving optimizations. • The definition via counit-unit adjunction is convenient for proofs about functors which are known to be adjoint, because they provide formulas that can be directly manipulated. • The definition via hom-sets makes symmetry the most apparent, and is the reason for using the word adjoint. Adjoint functors arise everywhere, in all areas of mathematics. Their full usefulness lies in that the structure in any of these definitions gives rise to the structures in the others via a long but trivial series of deductions. Thus, switching between them makes implicit use of a great deal of tedious details that would otherwise have to be repeated separately in every subject area. For example, naturality and terminality of the counit can be used to prove that any right adjoint functor preserves limits. Adjoint functors 307 A helpful writing convention The theory of adjoints has the terms left and right at its foundation, and there are many components which live in one of two categories C and D which are under consideration. It can therefore be extremely helpful to choose letters in alphabetical order according to whether they live in the "lefthand" category C or the "righthand" category D, and also to write them down in this order whenever possible. In this article for example, the letters X, F,f e will consistently denote things which live in the category C, the letters Y, G, g, r\ will consistently denote things which live in the category D, and whenever possible such things will be referred to in order from left to right (a functor F:C<^D can be thought of as "living" where its outputs are, in C). Definitions via universal morphisms A functor F : C <— D is a left adjoint functor if for each object X in C, there exists a terminal morphism from F to X. If, for each object X in C, we choose an object G X of D and a terminal morphism e : F(G X) — > X from F to X, then there is a unique functor G : C — > D such that GX = G X and e FG(f) =/e y for/ : X — > X' a morphism in C; F is then called a left adjoint to G. A functor G : C — > D is a right adjoint functor if for each object Y in D, there exists an initial morphism from Y to G. If, for each object Y in D, we choose an object FY of C and an initial morphism 11 : Y — > G(F Y) from Y to G, then there is a unique functor F : C <— D such that FY = FY and GF(g) f\ Y =f\ Y ,g for for g : Y — > Y' a morphism in D; G is then called a right adjoint to F. Remarks: It is true, as the terminology implies, that F is left adjoint to G if and only if G is right adjoint to F. This is apparent from the symmetric definitions given below. The definitions via universal morphisms are often useful for establishing that a given functor is left or right adjoint, because they are minimalistic in their requirements. They are also intuitively meaningful in that finding a universal morphism is like solving an optimization problem. Definition via counit-unit adjunction A counit-unit adjunction between two categories C and D consists of two functors F : C <— D and G : C — > D and two natural transformations e : FG -► l c T) : l v -> GF respectively called the counit and the unit of the adjunction (terminology from universal algebra*), such that the compositions f^Ufgf^f G^GFG-^G are the identity transformations 1 and 1 on F and G respectively. In this situation we say that F is left adjoint to G and G is right adjoint to F , and may indicate this relationship by writing (e y rj) : F H G , or simply F ~\ G ■ In equation form, the above conditions on (e,t|) are the counit-unit equations 1 F = eF o Ft] 1 g = Geo V G which mean that for each X in C and each Y in D, l FY = e FY o F(t) Y ) Igx = G{ex) ° Vgx Adjoint functors 308 These equations are useful in reducing proofs about adjoint functors to algebraic manipulations. They are sometimes called the zig-zag equations because of the appearance of the corresponding string diagrams. A way to remember them is to first write down the nonsensical equation 1 = £ o r\ and then fill in either F or G in one of the two simple ways which make the compositions defined. Note: The use of the prefix "co" in counit here is not consistent with the terminology of limits and colimits, because a colimit satisfies an initial property whereas the counit morphisms will satisfy terminal properties, and dually. The term unit here is borrowed from the theory of monads where it looks like the insertion of the identity 1 into a monoid. Definition via hom-set adjunction A hom-set adjunction between two categories C and D consists of two functors F : C <— D and G : C — > D and a natural isomorphism <& : liom c (F-, -) — s- hom fl (-, G-). This specifies a family of bijections ® Y , X : hom c (FY,X) -»■ hom D (Y, GX). for all objects X in C and Y in D. In this situation we say that F is left adjoint to G and G is right adjoint to F , and may indicate this relationship by writing <|> ; J? -| Q , or simply F ~\ G ■ This definition is a logical compromise in that it is somewhat more difficult to satisfy than the universal morphism definitions, and has fewer immediate implications than the counit-unit definition. It is useful because of its obvious symmetry, and as a stepping-stone between the other definitions. In order to interpret O as a natural isomorphism, one must recognize hom JF— , — ) and hom (— , G— ) as functors. In fact, they are both bifunctors from D op x C to Set (the category of sets). For details, see the article on hom functors. Explicitly, the naturality of O means that for all morphisms f: X following diagram commutes: X ' in C and all morphisms g : Y ' — > Y in D the Hom c (FY,X) Hom(FgJ) Kvm c [FY',X i ) <[> Y, X Kou^i Y,GX) Hom(3,G/) (|) Homp( Y* , GX 1 ) Y'X ! The vertical arrows in this diagram are those induced by composition with/and g. Adjoint functors 309 Adjunctions in full There are hence numerous functors and natural transformations associated with every adjunction, and only a small portion is sufficient to determine the rest. An adjunction between categories C and D consists of • A functor F : C <— D called the left adjoint • A functor G : C — > D called the right adjoint • A natural isomorphism O : horn (F— ,— ) hony-G-) > 1 called the counit GF called the unit • A natural transformation e : FG • A natural transformation T) : 1 - An equivalent formulation, where X denotes any object of C and Y denotes any object of D: For every C-morphism f : FY — > X there is a unique D-morphism ^Yx(f) — 9 '■ Y — * GX such that the diagrams below commute, and for every D-morphism g : Y — > GX there is a unique C-morphism <£>X y\9) = / ■ FY — > X in C such that the diagrams below commute: From this assertion, one can recover that: • The transformations e, r\, and O are related by the equations / = Q-^g) = e x o F(g) e hom c (F(Y), X) 9 = ® Y ,x(f) = G(f)or lY ehom D (y,G(X)) ^gxA^x) = £ x G hom c (FG(X),X) ®y,fy0-fy) =Vy G hom D (Y, GF(Y)) • The transformations e, 11 satisfy the counit-unit equations 1 F = eF o Ft] 1 g = Geo V G • Each pair (GX, £x) i s a terminal morphism from F to X in C • Each pair (FY, T}y)is> an initial morphism from Y to G in D In particular, the equations above allow one to define O, e, and r| in terms of any one of the three. However, the adjoint functors F and G alone are in general not sufficient to determine the adjunction. We will demonstrate the equivalence of these situations below. Adjoint functors 310 Universal morphisms induce hom-set adjunction Given a right adjoint functor G \ C > D m the sense of initial morphisms, we can construct a functor F : C < — D an d hom-set adjunction $ : iiom c (F-, -) — > horn^-, G-) in the following steps: • For each Y in D, choose an initial morphism OsTy,T7y) from Y to G, so we have rty — > G(Xy) ■ • Initiality of these morphisms allows us to construct a unique functor F : C ^ — D sucn that FY = Xy and rj : lp — > GF is a natural transformation. • For every morphism g : Y — ► GX , initiality of (FY,Tjy) means we can let ^yx{q) be the unique morphism f : FY —> X such that G(f) o Tfy — g ■ • The map |»yj : homc(FY, X} < — homxjfY", GX ) is injective by uniqueness and surjective because we can solve its defining equation for/. It is natural in X because 11 is natural, and natural in Y because G is a functor. Hence letting <&y,X = ^y^ : h.om.c(FY, X) — > h.orri£)(y, GX) gives a hom-set adjunction as required. A similar argument allows one to construct a hom-set adjunction from the terminal morphisms to a left adjoint functor. (The construction that starts with a right adjoint is slightly more common, since the right adjoint in many adjoint pairs is a trivially defined inclusion or forgetful functor.) Counit-unit adjunction induces hom-set adjunction Given functors F : C < — Z)> G : C — > D > an d a counit-unit adjunction (e y rj) : F H G , we can construct a hom-set adjunction $ : h.om D (F-, -) — ► hom^-, G-) in the following steps: • For each f : FY — > X and each g '. Y — ► GX , define ®Y,x(f) = G(f)°VY ^y,x{s) = e x o F(g) The transformations O and W are natural because 11 and e are natural. • Using, in order, that F is a functor, that e is natural, and the counit-unit equation 1 FY = £fy ° F(Vy} > we obtain mf = e x o FG(f) o Firiy) = / o e Fy o ^(77^) = fol FY = f hence 'PO is the identity transformation. • Dually, using that G is a functor, that 11 is natural, and the counit-unit equation \qx = G(ex) ° t]GX > we obtain 3>% = G(ejc) o GF( ff ) o 7/y = G(ejf ) o ?7 GX o g = lex ° 9 = 9 hence O'P is the identity transformation, so O is a natural isomorphism with inverse O" = I 5 . Adjoint functors 311 Hom-set adjunction induces all of the above Given functors F : C < D > G \ C >Z)> an d a hom-set adjunction <& : h.om.c(F — , — ) -^ hom^f— , G— ) , we can construct a counit-unit adjunction (e,r,):F^G, which defines families of initial and terminal morphisms, in the following steps: • Let e x = §- l xx (\ GX ) £ ~h.om c {FGX, X) for each X in C, where \ GX £ hom D (GX : GX) is the identity morphism. • Let Tfy — & y ,fy{1fy) £ h.om D (Y, GFY) for each Y in £>, where ljry G hom c (Fy, FF) is the identity morphism. • The bijectivity and naturality of O imply that each (GX, E x ) is a terminal morphism from X to F in C, and each (FY, Tfy) is an initial morphism from Y to G in D. • The naturality of O implies the naturality of e and T), and the two formulas M/) = G(/)o7|y ^i(j)=^oF(j) for each/: FF — > X and g: 7 — > GX (which completely determine O). • Substituting F7 for X and 77^ = <&yfyO-Fy) for g in the second formula gives the first counit-unit equation 1 FY = Efy ° F(rj Y ), and substituting GX for 7 and e x = ^GXl(-'-G-f) for/ in the first formula gives the second counit-unit equation Igx = G(e y ) O 77 G x • Historical perspective Ubiquity of adjoint functors The idea of an adjoint functor was formulated by Daniel Kan in 1958. Like many of the concepts in category theory, it was suggested by the needs of homological algebra, which was at the time devoted to computations. Those faced with giving tidy, systematic presentations of the subject would have noticed relations such as hom(F(X), Y) = hom(X, G(Y)) in the category of abelian groups, where F was the functor — (g) A (i- e - take the tensor product with A), and G was the functor hom(A,— ). The use of the equals sign is an abuse of notation; those two groups are not really identical but there is a way of identifying them that is natural. It can be seen to be natural on the basis, firstly, that these are two alternative descriptions of the bilinear mappings from X x A to Y. That is, however, something particular to the case of tensor product. In category theory the 'naturality' of the bijection is subsumed in the concept of a natural isomorphism. The terminology comes from the Hilbert space idea of adjoint operators T, U with <Tx,y> = <x,Uy>, which is formally similar to the above relation between hom-sets. We say that F is left adjoint to G, and G is right adjoint to F. Note that G may have itself a right adjoint that is quite different from F (see below for an example). The analogy to adjoint maps of Hilbert spaces can be made precise in certain contexts . If one starts looking for these adjoint pairs of functors, they turn out to be very common in abstract algebra, and elsewhere as well. The example section below provides evidence of this; furthermore, universal constructions, which may be more familiar to some, give rise to numerous adjoint pairs of functors. In accordance with the thinking of Saunders Mac Lane, any idea such as adjoint functors that occurs widely enough in mathematics should be studied for its own sake. Adjoint functors 312 Problems formulated with adjoint functors Mathematicians do not generally need the full adjoint functor concept. Concepts can be judged according to their use in solving problems, as well as for their use in building theories. The tension between these two motivations was especially great during the 1950s when category theory was initially developed. Enter Alexander Grothendieck, who used category theory to take compass bearings in other work — in functional analysis, homological algebra and finally algebraic geometry. It is probably wrong to say that he promoted the adjoint functor concept in isolation: but recognition of the role of adjunction was inherent in Grothendieck's approach. For example, one of his major achievements was the formulation of Serre duality in relative form — one could say loosely, in a continuous family of algebraic varieties. The entire proof turned on the existence of a right adjoint to a certain functor. This is something undeniably abstract, and non-constructive, but also powerful in its own way. The case of partial orders Every partially ordered set can be viewed as a category (with a single morphism between x and y if and only if x < y). A pair of adjoint functors between two partially ordered sets is called a Galois connection (or, if it is contravariant, an antitone Galois connection). See that article for a number of examples: the case of Galois theory of course is a leading one. Any Galois connection gives rise to closure operators and to inverse order-preserving bijections between the corresponding closed elements. As is the case for Galois groups, the real interest lies often in refining a correspondence to a duality (i.e. antitone order isomorphism). A treatment of Galois theory along these lines by Kaplansky was influential in the recognition of the general structure here. The partial order case collapses the adjunction definitions quite noticeably, but can provide several themes: • adjunctions may not be dualities or isomorphisms, but are candidates for upgrading to that status • closure operators may indicate the presence of adjunctions, as corresponding monads (cf. the Kuratowski closure axioms) • a very general comment of Martin Hyland is that syntax and semantics are adjoint: take C to be the set of all logical theories (axiomatizations), and D the power set of the set of all mathematical structures. For a theory T in C, let F(T) be the set of all structures that satisfy the axioms T; for a set of mathematical structures 5, let G(S) be the minimal axiomatization of S. We can then say that F(T) is a subset of S if and only if T logically implies G(S): the "semantics functor" F is left adjoint to the "syntax functor" G. • division is (in general) the attempt to invert multiplication, but many examples, such as the introduction of implication in propositional logic, or the ideal quotient for division by ring ideals, can be recognised as the attempt to provide an adjoint. Together these observations provide explanatory value all over mathematics. Examples Free groups (instructive example) The construction of free groups is an extremely common adjoint construction, and a useful example for making sense of the above details. Suppose that F : Grp <— Set is the functor assigning to each set Y the free group generated by the elements of Y, and that G : Grp — > Set is the forgetful functor functor, which assigns to each group X its underlying set. Then F is left adjoint to G: Terminal morphisms. For each group X, the group FGX is the free group generated freely by GX, the elements of X. Let e x : FGX — y X. be the group homomorphism which sends the generators of FGX to the elements of X Adjoint functors 313 they correspond to, which exists by the universal property of free groups. Then each (GX, Ex) i s a terminal morphism from F to X, because any group homomorphism from a free group FZ to X will factor through £ X ; FGX — > X via a unique sel from Z to GX. This means that (F,G) is an adjoint pair. Initial morphisms. For each set Y, the set GFY is just the underlying set of the free group FY generated by Y. Let T]y '■ Y — *■ GFY be the set map given by "inclusion of generators". Then each (FY, T}y) is an initial morphism from Y to G, because any set map from Y to the underlying set GW of a group will factor through T\y : Y — > GFY via a unique group homomorphism from FY to W. This also means that (F,G) is an adjoint pair. Hom-set adjunction. Maps from the free group FY to a group X correspond precisely to maps from the set Y to the set GX: each homomorphism from FY to X is fully determined by its action on generators. One can verify directly that this correspondence is a natural transformation, which means it is a hom-set adjunction for the pair (F,G). Counit-unit adjunction. One can also verify directly that e and 11 are natural. Then, a direct verification that they form a counit-unit adjunction (s rj) : F H G is as follows: The first counit-unit equation 1^- = eF o Ft) says that for each set Y the composition Fyl^FGFY-^FY should be the identity. The intermediate group FGFY is the free group generated freely by the words of the free group FY. (Think of these words as placed in parentheses to indicate that they are independent generators.) The arrow F{j)y) is the group homomorphism from FY into FGFY sending each generator y of FY to the corresponding word of length one (y) as a generator of FGFY. The arrow Efy is the group homomorphism from FGFY to FY sending each generator to the word of FY it corresponds to (so this map is "dropping parentheses"). The composition of these maps is indeed the identity on FY. The second counit-unit equation 1q = Ge o r\G says that for each group X the composition gx^Ugfgx^^gx should be the identity. The intermediate set GFGX is just the underlying set of FGX. The arrow T)gx is the "inclusion of generators" set map from the set GX to the set GFGX. The arrow G(ex) i s tne set map from GFGX to GX which underlies the group homomorphism sending each generator of FGX to the element of X it corresponds to ("dropping parentheses"). The composition of these maps is indeed the identity on GX. Free constructions and forgetful functors Free objects are all examples of a left adjoint to a forgetful functor which assigns to an algebraic object its underlying set. These algebraic free functors have generally the same description of as in the detailed description of the free group situation above. Diagonal functors and limits Products, fibred products, equalizers, and kernels are all examples of the categorical notion of a limit. Any limit functor is right adjoint to a corresponding diagonal functor (provided the category has the type of limits in question), and the counit of the adjunction provides the defining maps from the limit object. Below are some specific examples. • Products Let n : Grp — > Grp the functor which assigns to each pair (X , X ) the product group X xX , and let A : Grp <— Grp be the diagonal functor which assigns to every group X the pair (X, X) in the product category 2 Grp . The universal property of the product group shows that n is right-adjoint to A. The counit of this adjunction is the defining pair of projection maps from X xX to X and X which define the limit, and the unit is the diagonal inclusion of a group X into X xX (mapping x to (x,x)). The cartesian product of sets, the product of rings, the product of topological spaces etc. follow the same pattern; it can also be extended in a straightforward manner to more than just two factors. More generally, any Adjoint functors 314 type of limit is right adjoint to a diagonal functor. • Kernels. Consider the category D of homomorphisms of abelian groups. If/ : A — > B and/ : A — » B are two objects of D, then a morphism from/ to/ is a pair (g , g ) of morphisms such that g^ = f S A - Let G : D — » Ab be the functor which assigns to each homomorphism its kernel and let F : D <— Ab be the functor which maps the group A to the homomorphism A — > 0. Then G is right adjoint to F, which expresses the universal property of kernels. The counit of this adjunction is the defining embedding of a homomorphism's kernel into the homomorphism's domain, and the unit is the morphism identifying a group A with the kernel of the homomorphism A — > 0. A suitable variation of this example also shows that the kernel functors for vector spaces and for modules are right adjoints. Analogously, one can show that the cokernel functors for abelian groups, vector spaces and modules are left adjoints. Colimits and diagonal functors Coproducts, fibred coproducts, coequalizers, and cokernels are all examples of the categorical notion of a colimit. Any colimit functor is left adjoint to a corresponding diagonal functor (provided the category has the type of colimits in question), and the unit of the adjunction provides the defining maps into the colimit object. Below are some specific examples. 2 • Coproducts. If F : Ab <— Ab assigns to every pair (X , X ) of abelian groups their direct sum, and if G : Ab — > 2 12 Ab is the functor which assigns to every abelian group Y the pair (Y, Y), then F is left adjoint to G, again a consequence of the universal property of direct sums. The unit of this adjoint pair is the defining pair of inclusion maps from X and X into the direct sum, and the counit is the additive map from the direct sum of (X,X) to back to X (sending an element (a, b) of the direct sum to the element a+b of X). Analogous examples are given by the direct sum of vector spaces and modules, by the free product of groups and by the disjoint union of sets. Further examples In algebra • Adjoining an identity to a rng. This example was discussed in the motivation section above. Given a rng R, a multiplicative identity element can be added by taking RxZ and defining a Z-bilinear product with (r,0)(0,l) = (0,l)(r,0) = (r,0), (r,0)(s,0) = (rs,0), (0,1)(0,1) = (0,1). This constructs a left adjoint to the functor taking a ring to the underlying rng. • Ring extensions. Suppose R and S are rings, and p : R — » S is a ring homomorphism. Then S can be seen as a (left) 7?-module, and the tensor product with S yields a functor F : R -Mod — > S-Mod. Then F is left adjoint to the forgetful functor G : S-Mod — > fi-Mod. • Tensor products. If R is a ring and M is a right R module, then the tensor product with M yields a functor F : R-Mod — > Ab. The functor G : Ab — > 7?-Mod, defined by G(A) = hom (M,A) for every abelian group A, is a right adjoint to F. • From monoids and groups to rings The integral monoid ring construction gives a functor from monoids to rings. This functor is left adjoint to the functor that associates to a given ring its underlying multiplicative monoid. Similarly, the integral group ring construction yields a functor from groups to rings, left adjoint to the functor that assigns to a given ring its group of units. One can also start with a field K and consider the category of ^-algebras instead of the category of rings, to get the monoid and group rings over K. • Field of fractions. Consider the category Dom of integral domains with injective morphisms. The forgetful functor Field — > Dom from fields has a left adjoint - it assigns to every integral domain its field of fractions. Adjoint functors 315 • Polynomial rings. Let Ring^ be the category of pointed commutative rings with unity (pairs (A,a) where A is a ring, a £ A an d morphisms preserve the distinguished elements). The forgetful functor G:Ring + — > Ring has a left adjoint - it assigns to every ring R the pair (R[x],x) where R[x] is the polynomial ring with coefficients from R. • Abelianization. Consider the inclusion functor G : Ab — > Grp from the category of abelian groups to category of ab groups. It has a left adjoint called abelianization which assigns to every group G the quotient group G =G/[G,G]. • The Grothendieck group. In K-theory, the point of departure is to observe that the category of vector bundles on a topological space has a commutative monoid structure under direct sum. One may make an abelian group out of this monoid, the Grothendieck group, by formally adding an additive inverse for each bundle (or equivalence class). Alternatively one can observe that the functor that for each group takes the underlying monoid (ignoring inverses) has a left adjoint. This is a once-for-all construction, in line with the third section discussion above. That is, one can imitate the construction of negative numbers; but there is the other option of an existence theorem. For the case of finitary algebraic structures, the existence by itself can be referred to universal algebra, or model theory; naturally there is also a proof adapted to category theory, too. • Frobenius reciprocity in the representation theory of groups: see induced representation. This example foreshadowed the general theory by about half a century. In topology • A functor with a left and a right adjoint. Let G be the functor from topological spaces to sets that associates to every topological space its underlying set (forgetting the topology, that is). G has a left adjoint F, creating the discrete space on a set Y, and a right adjoint H creating the trivial topology on Y. • Suspensions and loop spaces Given topological spaces X and Y, the space [SX, Y] of homotopy classes of maps from the suspension SXof Xto Y is naturally isomorphic to the space [X, Q.Y] of homotopy classes of maps from X to the loop space Q.Y of Y. This is an important fact in homotopy theory. • Stone-Cech compactification. Let KHaus be the category of compact Hausdorff spaces and G : KHaus — > Top be the forgetful functor to the category of topological spaces. Then G has a left adjoint F : Top — > KHaus, the Stone— Cech compactification. The counit of this adjoint pair yields a continuous map from every topological space X into its Stone-Cech compactification. This map is an embedding (i.e. injective, continuous and open) if and only if X is a Tychonoff space. • Direct and inverse images of sheaves Every continuous map/ : X — > Y between topological spaces induces a functor/ from the category of sheaves (of sets, or abelian groups, or rings...) on X to the corresponding category of sheaves on Y, the direct image functor . It also induces a functor/ - from the category of sheaves of abelian groups on Y to the category of sheaves of abelian groups on X, the inverse image functor. f~ is left adjoint to/ . Here a more subtle point is that the left adjoint for coherent sheaves will differ from that for sheaves (of sets). • Soberification. The article on Stone duality describes an adjunction between the category of topological spaces and the category of sober spaces that is known as soberification. Notably, the article also contains a detailed description of another adjunction that prepares the way for the famous duality of sober spaces and spatial locales, exploited in pointless topology. Adjoint functors 316 In category theory • A series of adjunctions. The functor n which assigns to a category its sets of connected components is left-adjoint to the functor D which assigns to a set the discrete category on that set. Moreover, D is left-adjoint to the object functor U which assigns to each category its set of objects, and finally U is left-adjoint to A which assigns to each set the antidiscrete category on that set. • Exponential object. In a cartesian closed category the endofunctor C — > C given by — xA has a right adjoint — . In categorical logic • quantification Any morphism/ : X — > Y in a category with pullbacks induces a monotonous map /* : Subfin — > Sub (X) acting by pullbacks (A monotonous map is a functor if we consider the preorders as categories). If this functor has a left/right adjoint, the adjoint is called 3^ and \/j, respectively. In the category of sets, if we choose subsets as the canonical subobjects, then these functions are given by: (TCY) .-> f*(T) = f- 1 [T] (s c x) .-> 3 f x = { y e y | 3x e r'mU e s } = f[s] (S C X) .-> V f X = { y G Y \ Vx G /^[{y}],* G 5 } Properties Uniqueness of adjoints If the functor F : C <— D has two right adjoints G and G', then G and G' are naturally isomorphic. The same is true for left adjoints. Conversely, if F is left adjoint to G, and G is naturally isomorphic to G' then F is also left adjoint to G'. More generally, if (F, G, e, r\) is an adjunction (with counit-unit (e,r|)) and o : F -> f" t : G -> G' are natural isomorphisms then (F', G', e', T)') is an adjunction where 77' = (r * cr) o 7; £:' = e o (cr -1 * r _1 ). Here o denotes vertical composition of natural transformations, and * denotes horizontal composition. Composition Adjunctions can be composed in a natural fashion. Specifically, if (F, G, e, r)) is an adjunction between C and D and (F', G',e',r\') is an adjunction between D and E then the functor F'oF:C^S is left adjoint to GoG' :C^S. More precisely, there is an adjunction between F' F and G G' with unit and counit given by the compositions: l^GF^GG'FF F'FGG'^^F'G'^lc. This new adjunction is called the composition of the two given adjunctions. One can then form a category whose objects are all small categories and whose morphisms are adjunctions. Adjoint functors 317 Adjoints preserve limits The most important property of adjoints is their continuity: every functor that has a left adjoint (and therefore is a right adjoint) is continuous (i.e. commutes with limits in the category theoretical sense); every functor that has a right adjoint (and therefore is a left adjoint) is cocontinuous (i.e. commutes with colimits). Since many common constructions in mathematics are limits or colimits, this provides a wealth of information. For example: • applying a right adjoint functor to a product of objects yields the product of the images; • applying a left adjoint functor to a coproduct of objects yields the coproduct of the images; • every right adjoint functor is left exact; • every left adjoint functor is right exact. Additivity If C and D are preadditive categories and F : C <— D is an additive functor with a right adjoint G : C — > D, then G is also an additive functor and the hom-set bijections ® Y ,x ■ h.om c (FY, X) = h.om v (Y, GX) are, in fact, isomorphisms of abelian groups. Dually, if G is additive with a left adjoint F, then F is also additive. Moreover, if both C and D are additive categories (i.e. preadditive categories with all finite biproducts), then any pair of adjoint functors between them are automatically additive. General existence theorem Not every functor G : C — » D admits a left adjoint. If C is a complete category, then the functors with left adjoints can be characterized by the adjoint functor theorem of Peter J. Freyd: G has a left adjoint if and only if it is continuous and a certain smallness condition is satisfied: for every object 7 of D there exists a family of morphisms f.:Y^G(XJ where the indices i come from a set I, not a proper class, such that every morphism h : Y -» G(X) can be written as h = G(t) of. for some i in / and some morphism t : X -> X in C. i An analogous statement characterizes those functors with a right adjoint. Relationship to other categorical concepts Universal constructions As stated earlier, an adjunction between categories C and D gives rise to a family of universal morphisms, one for each object in C and one for each object in D. Conversely, if there exists a universal morphism to a functor G : C — > D from every object of D, then G has a left adjoint. However, universal constructions are more general than adjoint functors: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of D (equivalently, every object of C). Adjoint functors 318 Equivalences of categories If a functor F: C^>D is one half of an equivalence of categories then it is the left adjoint in an adjoint equivalence of categories, i.e. an adjunction whose unit and counit are isomorphisms. Every adjunction (F, G, e, r)) extends an equivalence of certain subcategories. Define C as the full subcategory of C consisting of those objects X of C for which e is an isomorphism, and define D as the full subcategory of D consisting of those objects 7 of D for which r| is an isomorphism. Then F and G can be restricted to D and C and yield inverse equivalences of these subcategories. In a sense, then, adjoints are "generalized" inverses. Note however that a right inverse of F (i.e. a functor G such that FG is naturally isomorphic to 1 ) need not be a right (or left) adjoint of F. Adjoints generalize two-sided inverses. Monads Every adjunction (F, G, e, r| ) gives rise to an associated monad ( T, r\, \i) in the category D. The functor T :V -^V is given by T = GF. The unit of the monad 77 : l p -s- T is just the unit 11 of the adjunction and the multiplication transformation fi, : T 2 -> T is given by u, = GeF. Dually, the triple (FG, £, Fi\G) defines a comonad in D. Every monad arises from some adjunction — in fact, typically from many adjunctions — in the above fashion. Two constructions, called the category of Eilenberg— Moore algebras and the Kleisli category are two extremal solutions to the problem of constructing an adjunction that gives rise to a given monad. References [1] arXiv.org: John C. Baez Higher-Dimensional Algebra II: 2-Hilbert Spaces (http://www.arxiv.org/abs/q-alg/9609018). • Adamek, Jiff; Horst Herrlich, and George E. Strecker (1990). Abstract and Concrete Categories (http://katmat. math.uni-bremen.de/acc/acc.pdf). John Wiley & Sons. ISBN 0-471-60922-6. • Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 ((2nd ed.) ed.). Springer- Verlag. ISBN 0-387-98403-8. External links • Adjunctions (http://www.youtube.com/view_play_list?p=54B49729E5 102248) Seven short lectures on adjunctions. Natural transformations 319 Natural transformations In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e. the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most basic notions of category theory and consequently appear in the majority of its applications. Definition Iff and G are functors between the categories C and D, then a natural transformation r| from F to G associates to every object X in C a morphism ri : F{X) — > G(X) in D called the component of r| at X, such that for every morphism/: X — > Y in C we have: Tjy o F(f) = G(f) o Vx This equation can conveniently be expressed by the commutative diagram F(X) — F(/) > F(Y) Vx VY If both F and G are contravariant, the horizontal arrows in this diagram are reversed. If 11 is a natural transformation from F to G, we also write 11 : F — > G or 11 : F => G. This is also expressed by saying the family of morphisms r\ : F(X) -> G(X) is natural in X. If, for every object X in C, the morphism r\ is an isomorphism in D, then 11 is said to be a natural isomorphism (or sometimes natural equivalence or isomorphism of functors). Two functors F and G are called naturally isomorphic or simply isomorphic if there exists a natural isomorphism from F to G. An infranatural transformation r\ from F to G is simply a family of morphisms 11 : F (X) — > G{X). Thus a natural transformation is an infranatural transformation for which ri o F(f) = G(f) o r| for every morphism/ : X — > Y. The naturalizer of n, nat(r|), is the largest subcategory of C containing all the objects of C on which r\ restricts to a natural transformation. Examples A worked example Statements such as "Every group is naturally isomorphic to its opposite group" abound in modern mathematics. We will now give the precise meaning of this statement as well as its proof. Consider the category Grp of all groups with group homomorphisms as morphisms. If (G,*) is a group, we define its opposite group (G°P ; * P) a s follows: G op is the same set as G, and the operation * op is defined by a* op b = b*a. All multiplications in G op are thus "turned around". Forming the opposite group becomes a (covariant!) functor from Grp to Grp if we define f p = / for any group homomorphism / G — > H. Note that f p is indeed a group homomorphism from G op to H° p : f p (a* op b) =f{b*a) =f{b)*f{a) =f p (a)* op f P (b). The content of the above statement is: Natural transformations 320 "The identity functor Id : Grp — > Grp is naturally isomorphic to the opposite functor op : Grp — > Grp." To prove this, we need to provide isomorphisms r| : G — > G op for every group G, such that the above diagram commutes. Set r\ Ja) = a . The formulas (ab)~ = b~ a and {a )' = a show that r\ is a group homomorphism which is its own inverse. To prove the naturality, we start with a group homomorphism/: G — > H and show il„o/ = f p o r) , i.e. (f{a))~ = f (a ) for all a in G. This is true since f p = f and every group homomorphism has the property if (a)) =f(d ). Further examples If K is a field, then for every vector space V over K we have a "natural" injective linear map V — > V** from the vector space into its double dual. These maps are "natural" in the following sense: the double dual operation is a functor, and the maps are the components of a natural transformation from the identity functor to the double dual functor. Every finite dimensional vector space is also isomorphic to its dual space. But this isomorphism relies on an arbitrary choice of basis, and is not natural, though there is an infranatural transformation. More generally, any vector spaces with the same dimensionality are isomorphic, but not naturally so. (Note however that if the space has a nondegenerate bilinear form, then there is a natural isomorphism between the space and its dual. Here the space is viewed as an object in the category of vector spaces and transposes of maps.) Consider the category Ab of abelian groups and group homomorphisms. For all abelian groups X, Y and Z we have a group isomorphism Hom(X®Y, Z) -> Hom(X, Hom(F, Z)). These isomorphisms are "natural" in the sense that they define a natural transformation between the two involved functors Ab x Ab op x Ab op — > Ab. Natural transformations arise frequently in conjunction with adjoint functors. Indeed, adjoint functors are defined by a certain natural isomorphism. Additionally, every pair of adjoint functors comes equipped with two natural transformations (generally not isomorphisms) called the unit and counit. Operations with natural transformations If i"| : F — > G and e : G — » H are natural transformations between functors F,G,H:C — > D, then we can compose them to get a natural transformation et) : F — > H. This is done componentwise: (er\) - e r\ . This "vertical composition" of natural transformation is associative and has an identity, and allows one to consider the collection of all functors C — » D itself as a category (see below under Functor categories). Natural transformations also have a "horizontal composition". If 11 : F — > G is a natural transformation between functors F,G:C — > D and e : / — > K is a natural transformation between functors /, K:D — > E, then the composition of functors allows a composition of natural transformations r|£ : JF — > KG. This operation is also associative with identity, and the identity coincides with that for vertical composition. The two operations are related by an identity which exchanges vertical composition with horizontal composition. If r| : F — > G is a natural transformation between functors F, G : C — > D, and H : D — > E is another functor, then we can form the natural transformation ffr| : HF — > HG by defining (Hri) x = Hri x . If on the other hand K : B — » C is a functor, the natural transformation r\K : FK — > GK is defined by (V K )x = Vk(x)- Natural transformations 321 Functor categories If C is any category and / is a small category, we can form the functor category C having as objects all functors from / to C and as morphisms the natural transformations between those functors. This forms a category since for any functor F there is an identity natural transformation 1 : F — > F (which assigns to every object X the identity morphism on F(X)) and the composition of two natural transformations (the "vertical composition" above) is again a natural transformation. The isomorphisms in C are precisely the natural isomorphisms. That is, a natural transformation 11 : F —> G is a natural isomorphism if and only if there exists a natural transformation e : G — » F such that rie = 1 and er\ = 1 . O F The functor category C is especially useful if / arises from a directed graph. For instance, if / is the category of the directed graph • — > *, then C has as objects the morphisms of C, and a morphism between cp : U — > V and op : X — » Y in C is a pair of morphisms/: U — > Xand g : V —> Yin C such that the "square commutes", i.e. tyf= g cp. More generally, one can build the 2-category Cat whose • 0-cells (objects) are the small categories, • 1 -cells (arrows) between two objects (J and £) are the functors from (J to £) , • 2-cells between two 1 -cells (functors) F \ G — > D an d G : C — > D are the natural transformations from JP toG- The horizontal and vertical compositions are the compositions between natural transformations described previously. A functor category (J 1 is then simply a hom-category in this category (smallness issues aside). Yoneda lemma If X is an object of a locally small category C, then the assignment Y h- > Horn (X, Y) defines a covariant functor F : C — > Set. This functor is called representable (more generally, a representable functor is any functor naturally isomorphic to this functor for an appropriate choice of X). The natural transformations from a representable functor to an arbitrary functor F : C — » Set are completely known and easy to describe; this is the content of the Yoneda lemma. Historical notes Saunders Mac Lane, one of the founders of category theory, is said to have remarked, "I didn't invent categories to study functors; I invented them to study natural transformations." Just as the study of groups is not complete without a study of homomorphisms, so the study of categories is not complete without the study of functors. The reason for Mac Lane's comment is that the study of functors is itself not complete without the study of natural transformations. The context of Mac Lane's remark was the axiomatic theory of homology. Different ways of constructing homology could be shown to coincide: for example in the case of a simplicial complex the groups defined directly, and those of the singular theory, would be isomorphic. What cannot easily be expressed without the language of natural transformations is how homology groups are compatible with morphisms between objects, and how two equivalent homology theories not only have the same homology groups, but also the same morphisms between those groups. References • Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 (2nd ed.). Springer- Verlag. ISBN 0-387-98403-8. Algebraic category 322 Algebraic category In mathematics, specifically universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature which is closed under the taking of homomorphic images, subalgebras and (direct) products. In the context of category theory, a variety of algebras is usually called a finitary algebraic category. A covariety is the class of all coalgebraic structures of a given signature. A variety of algebras should not be confused with an algebraic variety. Intuitively, a variety of algebras is an equationally defined collection of algebras, while an algebraic variety is an equationally defined collection of elements from a single algebra. The two are named alike by analogy, but they are formally quite distinct and their theories have little in common. Birkhoff's theorem Garrett Birkhoff proved equivalent the two definitions of variety given above, a result of fundamental importance to universal algebra and known as Birkhoff's theorem or as the HSP theorem. H, S, and P stand, respectively, for the closure operations of homomorphism, subalgebra, and product. An equational class for some signature 2 is the collection of all models, in the sense of model theory, that satisfy some set E of equations, asserting equality between terms. A model satisfies these equations if they are true in the model for any valuation of the variables. The equations in E are then said to be identities of the model. Examples of such identities are the commutative law, characterizing commutative algebras, and the absorption law, characterizing lattices. It is simple to see that the class of algebras satisfying some set of equations will be closed under the HSP operations. Proving the converse — classes of algebras closed under the HSP operations must be equational — is much harder. Examples The class of all semigroups forms a variety of algebras of signature (2). A sufficient defining equation is the associative law: x(yz) = (xy)z. It satisfies the HSP closure requirement, since any homomorphic image, any subset closed under multiplication and any direct product of semigroups is also a semigroup. The class of groups forms a class of algebras of signature (2,1,0), the three operations being respectively multiplication, inversion and identity. Any subset of a group closed under multiplication, under inversion and under identity (i.e. containing the identity) forms a subgroup. Likewise, the collection of groups is closed under homomorphic image and under direct product. Applying Birkhoff s theorem, this is sufficient to tell us that the groups form a variety, and so it should be defined by a collection of identities. In fact, the familiar axioms of associativity, inverse and identity form one suitable set of identities: x(yz) = (xy)z \x = x\ = X XX~ = X~ X = 1. A subvariety is a subclass of a variety, closed under the operations H, S, P. Notice that although every group is a semigroup, the class of groups does not form a subvariety of the variety of semigroups. This is because not every subsemigroup of a group is a group. Algebraic category 323 The class of abelian groups, considered again with signature (2,1,0), also has the HSP closure properties. It forms a subvariety of the variety of groups, and can be defined equationally by the three group axioms above together with the commutativity law: xy — yx. Variety of finite algebras Since varieties are closed under arbitrary cartesian products, all non-trivial varieties contain infinite algebras. It follows that the theory of varieties is of limited use in the study of finite algebras, where one must often apply techniques particular to the finite case. With this in mind, attempts have been made to develop a finitary analogue of the theory of varieties. A variety of finite algebras, sometimes called a pseudovariety, is usually defined to be a class of finite algebras of a given signature, closed under the taking of homomorphic images, subalgebras and finitary direct products. There is no general finitary counterpart to Birkhoff s theorem, but in many cases the introduction of a more complex notion of equations allows similar results to be derived. Pseudovarieties are of particular importance in the study of finite semigroups and hence in formal language theory. Eilenberg's theorem, often referred to as the variety theorem describes a natural correspondence between varieties of regular languages and pseudovarieties of finite semigroups. Category theory If A is a finitary algebraic category, then the forgetful functor U : A -> Set is monadic. Even more, it is strictly monadic, in that the comparison functor K : A -> Set T is an isomorphism (and not just an equivalence). Here, Set T i s tne Eilenberg— Moore category on Set- in general, one says a category is an algebraic category if it is monadic over Set ■ This is a more general notion than "finitary algebraic category" (the notion of "variety" used in universal algebra) because it admits such categories as CABA (complete atomic Boolean algebras) and CSLat (complete semilattices) whose signatures include infinitary operations. In those two cases the signature is large, meaning that it forms not a set but a proper class, because its operations are of unbounded arity. The algebraic category of sigma algebras also has infinitary operations, but their arity is countable whence its signature is small (forms a set). See also • Quasivariety Notes [1] Saunders Mac Lane, Categories for the Working Mathematician, Springer. (See p. 152) References Two monographs available free online: • Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. A Course in Universal Algebra. (http://www.thoralf. uwaterloo.ca/htdocs/ualg.html) Springer- Verlag. ISBN 3-540-90578-2. • Jipsen, Peter, and Henry Rose, 1992. Varieties of Lattices (http://wwwl.chapman.edu/~jipsen/ JipsenRoseVoL.html), Lecture Notes in Mathematics 1533. Springer Verlag. ISBN 0-387-56314-8. Domain theory 324 Domain theory Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational semantics, especially for functional programming languages. Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way and has close relations to topology. An alternative important approach to denotational semantics in computer science is that of metric spaces. Motivation and intuition The primary motivation for the study of domains, which was initiated by Dana Scott in the late 1960s, was the search for a denotational semantics of the lambda calculus. In this formalism, one considers "functions" specified by certain terms in the language. In a purely syntactic way, one can go from simple functions to functions that take other functions as their input arguments. Using again just the syntactic transformations available in this formalism, one can obtain so called fixed point combinators (the best-known of which is the Y combinator); these, by definition, have the property that/(Y(/)) = Y(f) for all functions/. To formulate such a denotational semantics, one might first try to construct a model for the lambda calculus, in which a genuine (total) function is associated with each lambda term. Such a model would formalize a link between the lambda calculus as a purely syntactic system and the lambda calculus as a notational system for manipulating concrete mathematical functions. Unfortunately, such a model cannot exist, for if it did, it would have to contain a genuine, total function that corresponds to the combinator Y, that is, a function that computes a fixed point of an arbitrary input function/. There can be no such function for Y, because some functions (for example, the successor function) do not have a fixed point. At best, the genuine function corresponding to Y would have to be a partial function, necessarily undefined at some inputs. Scott got around this difficulty by formalizing a notion of "partial" or "incomplete" information to represent computations that have not yet returned a result. This was modeled by considering, for each domain of computation (e.g. the natural numbers), an additional element that represents an undefined output, i.e. the "result" of a computation that never ends. In addition, the domain of computation is equipped with an ordering relation, in which the "undefined result" is the least element. The important step to find a model for the lambda calculus is to consider only those functions (on such a partially ordered set) which are guaranteed to have least fixed points. The set of these functions, together with an appropriate ordering, is again a "domain" in the sense of the theory. But the restriction to a subset of all available functions has another great benefit: it is possible to obtain domains that contain their own function spaces, i.e. one gets functions that can be applied to themselves. Beside these desirable properties, domain theory also allows for an appealing intuitive interpretation. As mentioned above, the domains of computation are always partially ordered. This ordering represents a hierarchy of information or knowledge. The higher an element is within the order, the more specific it is and the more information it contains. Lower elements represent incomplete knowledge or intermediate results. Computation then is modeled by applying monotone functions repeatedly on elements of the domain in order to refine a result. Reaching a fixed point is equivalent to finishing a calculation. Domains provide a superior setting for these ideas since fixed points of monotone functions can be guaranteed to exist and, under additional restrictions, can be approximated from below. Domain theory 325 A guide to the formal definitions In this section, the central concepts and definitions of domain theory will be introduced. The above intuition of domains being information orderings will be emphasized to motivate the mathematical formalization of the theory. The precise formal definitions are to be found in the dedicated articles for each concept. A list of general order-theoretic definitions which include domain theoretic notions as well can be found in the order theory glossary. The most important concepts of domain theory will nonetheless be introduced below. Directed sets as converging specifications As mentioned before, domain theory deals with partially ordered sets to model a domain of computation. The goal is to interpret the elements of such an order as pieces of information or (partial) results of a computation, where elements that are higher in the order extend the information of the elements below them in a consistent way. From this simple intuition it is already clear that domains often do not have a greatest element, since this would mean that there is an element that contains the information of all other elements - a rather uninteresting situation. A concept that plays an important role in the theory is the one of a directed subset of a domain, i.e. of a non-empty subset of the order in which each two elements have some upper bound that is an element of this subset. In view of our intuition about domains, this means that every two pieces of information within the directed subset are consistently extended by some other element in the subset. Hence we can view directed sets as consistent specifications, i.e. as sets of partial results in which no two elements are contradictory. This interpretation can be compared with the notion of a convergent sequence in analysis, where each element is more specific than the preceding one. Indeed, in the theory of metric spaces, sequences play a role that is in many aspects analogous to the role of directed sets in domain theory. Now, as in the case of sequences, we are interested in the limit of a directed set. According to what was said above, this would be an element that is the most general piece of information which extends the information of all elements of the directed set, i.e. the unique element that contains exactly the information that was present in the directed set - and nothing more. In the formalization of order theory, this is just the least upper bound of the directed set. As in the case of limits of sequences, least upper bounds of directed sets do not always exist. Naturally, one has a special interest in those domains of computations in which all consistent specifications converge, i.e. in orders in which all directed sets have a least upper bound. This property defines the class of directed complete partial orders, or dcpo for short. Indeed, most considerations of domain theory do only consider orders that are at least directed complete. From the underlying idea of partially specified results as representing incomplete knowledge, one derives another desirable property: the existence of a least element. Such an element models that state of no information - the place where most computations start. It also can be regarded as the output of a computation that does not return any result at all. Computations and domains Now that we have some basic formal descriptions of what a domain of computation should be, we can turn to the computations themselves. Clearly, these have to be functions, taking inputs from some computational domain and returning outputs in some (possibly different) domain. However, one would also expect that the output of a function will contain more information when the information content of the input is increased. Formally, this means that we want a function to be monotonic. When dealing with dcpos, one might also want computations to be compatible with the formation of limits of a directed set. Formally, this means that, for some function /, the image f{D) of a directed set D (i.e. the set of the images of each element of D) is again directed and has as a least upper bound the image of the least upper bound of D. One could also say that / preserves directed suprema. Also note that, by considering directed sets of two Domain theory 326 elements, such a function also has to be monotonic. These properties give rise to the notion of a Scott-continuous function. Since this often is not ambiguous one also may speak of continuous functions. Approximation and finiteness Domain theory is a purely qualitative approach to modeling the structure of information states. One can say that something contains more information, but the amount of additional information is not specified. Yet, there are some situations in which one wants to speak about elements that are in a sense much simpler (or much more incomplete) than a given state of information. For example, in the natural subset-inclusion ordering on some powerset, any infinite element (i.e. set) is much more "informative" than any of its finite subsets. If one wants to model such a relationship, one may first want to consider the induced strict order < of a domain with order <. However, while this is a useful notion in the case of total orders, it does not tell us much in the case of partially ordered sets. Considering again inclusion-orders of sets, a set is already strictly smaller than another, possibly infinite, set if it contains just one less element. One would, however, hardly agree that this captures the notion of being "much simpler". Way-below relation A more elaborate approach leads to the definition of the so-called order of approximation, which is more suggestively also called the way-below relation. An element x is way below an element y, if, for every directed set D with supremum such that y < sup D, there is some element d in D such that x < d. Then one also says that x approximates y and writes x « y. This does imply that x < y, since the singleton set {y} is directed. For an example, in an ordering of sets, an infinite set is way above any of its finite subsets. On the other hand, consider the directed set (in fact: the chain) of finite sets {0}, {0,1}, {0,1,2},... Since the supremum of this chain is the set of all natural numbers N, this shows that no infinite set is way below N. However, being way below some element is a relative notion and does not reveal much about an element alone. For example, one would like to characterize finite sets in an order-theoretic way, but even infinite sets can be way below some other set. The special property of these finite elements x is that they are way below themselves, i.e. x « x. An element with this property is also called compact. Yet, such elements do not have to be "finite" nor "compact" in any other mathematical usage of the terms. The notation is nonetheless motivated by certain parallels to the respective notions in set theory and topology. The compact elements of a domain have the important special property that they cannot be obtained as a limit of a directed set in which they did not already occur. Many other important results about the way-below relation support the claim that this definition is appropriate to capture many important aspects of a domain. Domain theory 327 Bases of domains The previous thoughts raise another question: is it possible to guarantee that all elements of a domain can be obtained as a limit of much simpler elements? This is quite relevant in practice, since we cannot compute infinite objects but we may still hope to approximate them arbitrarily closely. More generally, we would like to restrict to a certain subset of elements as being sufficient for getting all other elements as least upper bounds. Hence, one defines a base of a poset P as being a subset B of P, such that, for each x in P, the set of elements in B that are way below x contains a directed set with supremum x. The poset P is a continuous poset if it has some base. Especially, P itself is a base in this situation. In many applications, one restricts to continuous (d)cpos as a main object of study. Finally, an even stronger restriction on a partially ordered set is given by requiring the existence of a base of compact elements. Such a poset is called algebraic. From the viewpoint of denotational semantics, algebraic posets are particularly well-behaved, since they allow for the approximation of all elements even when restricting to finite ones. As remarked before, not every finite element is "finite" in a classical sense and it may well be that the finite elements constitute an uncountable set. In some cases, however, the base for a poset is countable. In this case, one speaks of an to-continuous poset. Accordingly, if the countable base consists entirely of finite elements, we obtain an order that is to-algebraic. Special types of domains A simple special case of a domain is known as an elementary or flat domain. This consists of a set of incomparable elements, such as the integers, along with a single "bottom" element considered smaller than all other elements. One can obtain a number of other interesting special classes of ordered structures that could be suitable as "domains". We already mentioned continuous posets and algebraic posets. More special versions of both are continuous and algebraic epos. Adding even further completeness properties one obtains continuous lattices and algebraic lattices, which are just complete lattices with the respective properties. For the algebraic case, one finds broader classes of posets which are still worth studying: historically, the Scott domains were the first structures to be studied in domain theory. Still wider classes of domains are constituted by SFP -domains, L-domains, and bifinite domains. All of these classes of orders can be cast into various categories of depos, using functions which are monotone, Scott-continuous, or even more specialized as morphisms. Finally, note that the term domain itself is not exact and thus is only used as an abbreviation when a formal definition has been given before or when the details are irrelevant. Important results A poset D is a depo if and only if each chain in D has a supremum. However, directed sets are strictly more powerful than chains. If/ is a continuous function on a poset D then it has a least fixed point, given as the least upper bound of all finite iterations of fon the least element 0: V . ^ T f n (0). J ninN Generalizations • Synthetic domain theory • Topological domain theory • A continuity space is a generalization of metric spaces and posets, that can be used to unify the notions of metric spaces and domains. Domain theory 328 Further reading • G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott (2003). "Continuous Lattices and Domains". Encyclopedia of Mathematics and its Applications . 93. Cambridge University Press. ISBN 0-521-80338-1. • S. Abramsky, A. Jung (1994). "Domain theory" . In S. Abramsky, D. M. Gabbay, T. S. E. Maibaum, editors, (PDF). Handbook of Logic in Computer Science. III. Oxford University Press. ISBN 0-19-853762-X. Retrieved 2007-10-13. mi • Alex Simpson (2001-2002). "Part III: Topological Spaces from a Computational Perspective" . Mathematical Structures for Semantics. Retrieved 2007-10-13. • D. S. Scott (1975). "Data types as lattices". Proceedings of the International Summer Institute and Logic Colloquium, Kiel, in Lecture Notes in Mathematics (Springer- Verlag) 499: 579—651. • Carl A. Gunter (1992). Semantics of Programming Languages. MIT Press. • B. A. Davey and H. A. Priestley (2002). Introduction to Lattices and Order (2nd ed.). Cambridge University Press. ISBN 0-521-78451-4. • Carl Hewitt and Henry Baker (August 1977). "Actors and Continuous Functionals". Proceedings oflFIP Working Conference on Formal Description of Programming Concepts. References [1] http://citeseerx.ist. psu.edu/viewdoc/download?doi=10. 1. 1.55.903&rep=repl&type=pdf [2] http://homepages.inf.ed.ac.uk/als/Research/topological-domain-theory.html [3] http://www.cs.bham.ac.uk/~axj/pub/papers/handyl.pdf [4] http://www.dcs.ed.ac.uk/home/als/Teaching/MSfS/13.ps Enriched category theory In category theory and its applications to mathematics, an enriched category is a category whose hom-sets are replaced by objects from some other category, in a well-behaved manner. Definition We define here what it means for C to be an enriched category over a monoidal category (M, ® , I) ■ The following structures are required: • Let Ob(C) be a set (or proper class). An element of Ob(C) is called an object of C. • For each pair (A,B) of objects of C, let Hom(A,B) be an object of M, called the horn-object of A and B. • For each object A of C, let id be a morphism in M from / to Hom(A,A), called the identity morphism of A. • For each triple (A,B,C) of objects of C, let o : Hom(B, C) ® Hom(^, B) -> Hom(A, C) be a morphism in M called the composition morphism of A, B, and C. The following axioms are required: • Associativity: Given objects A, B, C, and D of C, we can go from Hom(C,D) <E> Hom(B,C) <E> Hom(A,B) to Hom(A,D) in two ways, depending on which composition we do first. These must give the same result. (Hom(C\ D) (S Hom(S, C)) ® Kam(A, B) -r- HomfC, D) (3 (HomfB, C) (S Honi(A, B)) *- Hom(C, D) !g> Homf.-l, C) HomlB, D) 3 Hom(A, E) ^- Hom(/l, D) Enriched category theory 329 • Left identity: Given objects A and B of C, we can go from / <E> Hom(A,B) to just Hom(A,B) in two ways, either by using id on I and then using composition, or by simply using the fact that I is an identity for <E> in M. These must give the same result. • Right identity: Given objects A and B of C, we can go from Hom(A,B) <E> / to just Hom(A,B) in two ways, either by using id on I and then using composition, or by simply using the fact that I is an identity for <E> in M. These must give the same result. Given the above, C (consisting of all the structures listed above) is a category enriched over M. Examples The most straightforward example is to take M to be a category of sets, with the Cartesian product for the monoidal operation. Then C is nothing but an ordinary category. If M is the category of small sets, then C is a locally small category, because the hom-sets will all be small. Similarly, if M is the category of finite sets, then C is a locally finite category. If M is the category 2 with Ob(2) = {0,1 }, a single nonidentity morphism (from to 1), and ordinary multiplication of numbers as the monoidal operation, then C can be interpreted as a preordered set. Specifically, A < B iff Hom(A,B) = l. If M is a category of pointed sets with smash product for the monoidal operation, then C is a category with zero morphisms. Specifically, the zero morphism from A to B is the special point in the pointed set Hom(A,B). If M is a category of abelian groups with tensor product as the monoidal operation, then C is a preadditive category. Relationship with monoidal functors If there is a monoidal functor from a monoidal category M to a monoidal category N, then any category enriched over M can be reinterpreted as a category enriched over N. Every monoidal category M has a monoidal functor M(7, — ) to the category of sets, so any enriched category has an underlying ordinary category. In many examples (such as those above) this functor is faithful, so a category enriched over M can be described as an ordinary category with certain additional structure or properties. Enriched functors An enriched functor is the appropriate generalization of the notion of a functor to enriched categories. Enriched functors are then maps between enriched categories which respect the enriched structure. If C and D are M-categories (that is, categories enriched over monoidal category M), an M-enriched functor T: C — > D is a map which assigns to each object of C an object of D and for each pair of objects a and b in C provides a morphism in M T : C{a,b) — > D(T(a),T(b)) between the hom-objects of C and D (which are objects in M), satisfying enriched versions of the axioms of a functor, viz preservation of identity and composition. Because the hom-objects need not be sets in an enriched category, one cannot speak of a particular morphism. There is no longer any notion of an identity morphism, nor of a particular composition of two morphisms. Instead, morphisms from the unit to a hom-object should be thought of as selecting an identity and morphisms from the monoidal product should be thought of as composition. The usual functorial axioms are replaced with corresponding commutative diagrams involving these morphisms. In detail, one has that the diagram Enriched category theory 330 commutes, which amounts to the equation T aa o id Q = id r(a ) , where / is the unit object of M. This is analogous to the rule F(id ) = id for ordinary functors. Additionally, one demands that the diagram C(b, c) ® C(a, b) D(T(fr),r(c))®D(r(«),T(6)) C{a,c) <-AC £»(T(a),T(c)) commute, which is analogous to the rule F(fg)=F(f)F(g) for ordinary functors. References ,M [1] Kelly, G.M. "Basic Concepts of Enriched Category Theory" , London Mathematical Society Lecture Note Series No.64 (C.U.P., 1982) References [1] http://www.tac.mta.ca/tac/reprints/articles/10/trl0.pdf Topos 331 Topos In mathematics, a topos (plural "topoi" or "toposes") is a type of category that behaves like the category of sheaves of sets on a topological space. For a discussion of the history of topos theory, see the article Background and genesis of topos theory. Grothendieck topoi (topoi in geometry) Since the introduction of sheaves into mathematics in the 1940s a major theme has been to study a space by studying sheaves on a space. This idea was expounded by Alexander Grothendieck by introducing the notion of a topos. The main utility of this notion is in the abundance of situations in mathematics where topological intuition is very effective but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the intuition. The greatest single success of this programmatic idea to date has been the introduction of the etale topos of a scheme. Equivalent formulations Let C be a category. A theorem of Giraud states that the following are equivalent: • There is a small category D and an inclusion C c —> Presh(D) that admits a finite-limit-preserving left adjoint. • C is the category of sheaves on a Grothendieck site. • C satisfies Giraud's axioms, below. A category with these properties is called a "(Grothendieck) topos". Here Presh(D) denotes the category of contravariant functors from D to the category of sets; such a contravariant functor is frequently called a presheaf. Giraud's axioms Giraud's axioms for a category C are: • C has a small set of generators, and admits all small colimits. Furthermore, colimits commute with fiber products. • Sums in C are disjoint. In other words, the fiber product of X and Y over their sum is the initial object in C. • All equivalence relations in C are effective. The last axiom needs the most explanation. If X is an object of C, an equivalence relation R on X is a map R^XxX in C such that all the maps Hom(Y,R)^Hom(Y,X)xHom(Y,X) are equivalence relations of sets. Since C has colimits we may form the coequalizer of the two maps R^X; call this XIR. The equivalence relation is effective if the canonical map R — > X x x / R X is an isomorphism. Examples Giraud's theorem already gives "sheaves on sites" as a complete list of examples. Note, however, that nonequivalent sites often give rise to equivalent topoi. As indicated in the introduction, sheaves on ordinary topological spaces motivate many of the basic definitions and results of topos theory. The category of sets is an important special case: it plays the role of a point in topos theory. Indeed, a set may be thought of as a sheaf on a point. More exotic examples, and the raison d'etre of topos theory, come from algebraic geometry. To a scheme and even a stack one may associate an etale topos, an fppf topos, a Nisnevich topos... Topos 332 Counterexamples Topos theory is, in some sense, a generalization of classical point-set topology. One should therefore expect to see old and new instances of pathological behavior. For instance, there is an example due to Pierre Deligne of a nontrivial topos that has no points (see below). Geometric morphisms If X and Y are topoi, a geometric morphism u: X— »F is a pair of adjoint functors (u ,u ) such that u preserves finite limits. Note that u automatically preserves colimits by virtue of having a right adjoint. By Freyd's adjoint functor theorem, to give a geometric morphism X — > Y is to give a functor u : Y — > X that preserves finite limits and all small colimits. Thus geometric morphisms between topoi may be seen as analogues of maps of locales. If X and Y are topological spaces and u is a continuous map between them, then the pullback and pushforward operations on sheaves yield a geometric morphism between the associated topoi. Points of topoi A point of a topos X is a geometric morphism from the topos of sets to X. If X is an ordinary space and x is a point of X, then the functor that takes a sheaf F to its stalk F has a right adjoint (the "skyscraper sheaf" functor), so an ordinary point of X also determines a topos-theoretic point. These may be constructed as the pullback-pushforward along the continuous map x: 1 — > X. Essential geometric morphisms A geometric morphism (u ,u ) is essential if u has a further left adjoint w , or equivalently (by the adjoint functor theorem) if u preserves not only finite but all small limits. Ringed topoi A ringed topos is a pair (X,R), where X is a topos and R is a commutative ring object in X. Most of the constructions of ringed spaces go through for ringed topoi. The category of 7?-module objects in X is an abelian category with enough injectives. A more useful abelian category is the subcategory of quasi-coherent 7?-modules: these are 7?-modules that admit a presentation. Another important class of ringed topoi, besides ringed spaces, are the etale topoi of Deligne-Mumford stacks. Homotopy theory of topoi Michael Artin and Barry Mazur associated to any topos a pro-simplicial set. Using this inverse system of simplicial sets one may sometimes associate to a homotopy invariant in classical topology an inverse system of invariants in topos theory. The pro-simplicial set associated to the etale topos of a scheme is a pro-finite simplicial set. Its study is called etale homotopy theory. Elementary toposes (toposes in logic) Introduction A traditional axiomatic foundation of mathematics is set theory, in which all mathematical objects are ultimately represented by sets (even functions which map between sets). More recent work in category theory allows this foundation to be generalized using toposes; each topos completely defines its own mathematical framework. The category of sets forms a familiar topos, and working within this topos is equivalent to using traditional set theoretic Topos 333 mathematics. But one could instead choose to work with many alternative toposes. A standard formulation of the axiom of choice makes sense in any topos, and there are toposes in which it is invalid. Constructivists will be interested to work in a topos without the law of excluded middle. If symmetry under a particular group G is of importance, one can use the topos consisting of all G-sets. It is also possible to encode an algebraic theory, such as the theory of groups, as a topos. The individual models of the theory, i.e. the groups in our example, then correspond to functors from the encoding topos to the category of sets that respect the topos structure. Formal definition When used for foundational work a topos will be defined axiomatically; set theory is then treated as a special case of topos theory. Building from category theory, there are multiple equivalent definitions of a topos. The following has the virtue of being concise, if not illuminating: A topos is a category which has the following two properties: • All limits taken over finite index categories exist. • Every object has a power object. From this one can derive that • All colimits taken over finite index categories exist. • The category has a subobject classifier. • Any two objects have an exponential object. • The category is cartesian closed. In many applications, the role of the subobject classifier is pivotal, whereas power objects are not. Thus some definitions reverse the roles of what is defined and what is derived. Explanation A topos as defined above can be understood as a cartesian closed category for which the notion of subobject of an object has an elementary or first-order definition. This notion, as a natural categorical abstraction of the notions of subset of a set, subgroup of a group, and more generally subalgebra of any algebraic structure, predates the notion of topos. It is definable in any category, not just toposes, in second-order language, i.e. in terms of classes of morphisms instead of individual morphisms, as follows. Given two monies m, n from respectively Y and Z to X, we say that m < n when there exists a morphism p: Y — > Z for which np = m, inducing a preorder on monies to X. When m < n and n < m we say that m and n are equivalent. The subobjects of X are the resulting equivalence classes of the monies to it. In a topos "subobject" becomes, at least implicitly, a first-order notion, as follows. As noted above, a topos is a category C having all finite limits and hence in particular the empty limit or final object 1 . It is then natural to treat morphisms of the form x: 1 — > X as elements x£X. Morphisms f.X^Y thus correspond to functions mapping each element x € X to the element fx € Y, with application realized by composition. One might then think to define a subobject of X as an equivalence class of monies m: X' — > X having the same image or range {mx\ x G X' }. The catch is that two or more morphisms may correspond to the same function, that is, we cannot assume that C is concrete in the sense that the functor C(l,-): C — > Set is faithful. For example the category Grph of graphs and their associated homomorphisms is a topos whose final object 1 is the graph with one vertex and one edge (a self-loop), but is not concrete because the elements 1 — > G of a graph G correspond only to the self-loops and not the other edges, nor the vertices without self-loops. Whereas the second-order definition makes G and its set of self-loops (with their vertices) distinct subobjects of G (unless every edge is, and every vertex has, a self-loop), this image-based one does not. This can be addressed for the graph example and related examples via the Yoneda Lemma as described in the Examples section below, but this then ceases to be first-order. Toposes provide a more Topos 334 abstract, general, and first-order solution. As noted above a topos C has a subobject classifier £2, namely an object of C with an element t € Q., the generic subobject of C, having the property that every monic m: X' — > X arises as a pullback of the generic subobject along a unique morphism/: X — » £2, as per Figure 1. Now the pullback of a monic is a monic, and all elements including t are monies since there is only one morphism to 1 from any given object, whence the pullback of t along /: X — > Q. is a monic. The monies to X are therefore in bijection with the pullbacks of t along morphisms from X to Q.. The latter morphisms partition the monies into equivalence classes each determined by a morphism /: X — > Q, the characteristic morphism of that class, which we take to be the subobject of X characterized or named by/. Figure I. mas a pullback of the generic subobject t along/. All this applies to any topos, whether or not concrete. In the concrete case, namely C(l,-) faithful, for example the category of sets, the situation reduces to the familiar behavior of functions. Here the monies m: X' — > X are exactly the injections (one-one functions) from X' to X, and those with a given image {mx\ x € X' } constitute the subobject of X corresponding to the morphism/: X — > Q. for which/" (?) is that image. The monies of a subobject will in general have many domains, all of which however will be in bijection with each other. To summarize, this first-order notion of subobject classifier implicitly defines for a topos the same equivalence relation on monies to X as had previously been defined explicitly by the second-order notion of subobject for any category. The notion of equivalence relation on a class of morphisms is itself intrinsically second-order, which the definition of topos neatly sidesteps by explicitly defining only the notion of subobject classifier Q., leaving the notion of subobject of X as an implicit consequence characterized (and hence namable) by its associated morphism/ X — > Further examples If C is a small category, then the functor category Set (consisting of all covariant functors from C to sets, with natural transformations as morphisms) is a topos. For instance, the category Grph of graphs of the kind permitting multiple directed edges between two vertices is a topos. A graph consists of two sets, an edge set and a vertex set, and two functions s,t between those sets, assigning to every edge e its source s(e) and target t{e). Grph is thus equivalent to the functor category Set , where C is the category with two objects E and V and two morphisms s,t: E — > V giving respectively the source and target of each edge. The categories of finite sets, of finite G-sets (actions of a group G on a finite set), and of finite graphs are also toposes. The Yoneda Lemma asserts that C op embeds in Set as a full subcategory. In the graph example the embedding represents C op as the subcategory of Set whose two objects are V as the one-vertex no-edge graph and E' as the two-vertex one-edge graph (both as functors), and whose two nonidentity morphisms are the two graph homomorphisms from V to E' (both as natural transformations). The natural transformations from V to an arbitrary graph (functor) G constitute the vertices of G while those from E' to G constitute its edges. Although Set , which we 2 can identify with Grph, is not made concrete by either V or E' alone, the functor U: Grph — > Set sending object G to the pair of sets (Grph(V,G), Grph(£',G)) and morphism h: G — > H to the pair of functions (Grph(V',/i), Grph(£',/z)) is faithful. That is, a morphism of graphs can be understood as a pair of functions, one mapping the vertices and the other the edges, with application still realized as composition but now with multiple sorts of generalized elements. This shows that the traditional concept of a concrete category as one whose objects have an underlying set can be generalized to cater for a wider range of toposes by allowing an object to have multiple Topos 335 underlying sets, that is, to be multisorted. References Some gentle papers • John Baez: "Topos theory in a nutshell. A gentle introduction. • Steven Vickers: "Toposes pour les nuls and "Toposes pour les vraiment nuls. Elementary and even more elementary introductions to toposes as generalized spaces. • Illusie, Luc, "What is a ... topos?" [ , Notices of the AMS The following texts are easy-paced introductions to toposes and the basics of category theory. They should be suitable for those knowing little mathematical logic and set theory, even non-mathematicians. • F. Willi